solving word problems with inequalities

Solving Inequality Word Questions

(You might like to read Introduction to Inequalities and Solving Inequalities first.)

In Algebra we have "inequality" questions like:

Sam and Alex play in the same soccer team. Last Saturday Alex scored 3 more goals than Sam, but together they scored less than 9 goals. What are the possible number of goals Alex scored?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

• Read the whole thing first
• Do a sketch if needed
• Assign letters for the values
• Find or work out formulas

We should also write down what is actually being asked for , so we know where we are going and when we have arrived!

The best way to learn this is by example, so let's try our first example:

Assign Letters:

• the number of goals Alex scored: A
• the number of goals Sam scored: S

We know that Alex scored 3 more goals than Sam did, so: A = S + 3

And we know that together they scored less than 9 goals: S + A < 9

We are being asked for how many goals Alex might have scored: A

Sam scored less than 3 goals, which means that Sam could have scored 0, 1 or 2 goals.

Alex scored 3 more goals than Sam did, so Alex could have scored 3, 4, or 5 goals .

• When S = 0, then A = 3 and S + A = 3, and 3 < 9 is correct
• When S = 1, then A = 4 and S + A = 5, and 5 < 9 is correct
• When S = 2, then A = 5 and S + A = 7, and 7 < 9 is correct
• (But when S = 3, then A = 6 and S + A = 9, and 9 < 9 is incorrect)

Lots More Examples!

Example: Of 8 pups, there are more girls than boys. How many girl pups could there be?

• the number of girls: g
• the number of boys: b

We know that there are 8 pups, so: g + b = 8, which can be rearranged to

We also know there are more girls than boys, so:

We are being asked for the number of girl pups: g

So there could be 5, 6, 7 or 8 girl pups.

Could there be 8 girl pups? Then there would be no boys at all, and the question isn't clear on that point (sometimes questions are like that).

• When g = 8, then b = 0 and g > b is correct (but is b = 0 allowed?)
• When g = 7, then b = 1 and g > b is correct
• When g = 6, then b = 2 and g > b is correct
• When g = 5, then b = 3 and g > b is correct
• (But if g = 4, then b = 4 and g > b is incorrect)

A speedy example:

Example: Joe enters a race where he has to cycle and run. He cycles a distance of 25 km, and then runs for 20 km. His average running speed is half of his average cycling speed. Joe completes the race in less than 2½ hours, what can we say about his average speeds?

• Average running speed: s
• So average cycling speed: 2s
• Speed = Distance Time
• Which can be rearranged to: Time = Distance Speed

We are being asked for his average speeds: s and 2s

The race is divided into two parts:

• Distance = 25 km
• Average speed = 2s km/h
• So Time = Distance Average Speed = 25 2s hours
• Distance = 20 km
• Average speed = s km/h
• So Time = Distance Average Speed = 20 s hours

Joe completes the race in less than 2½ hours

• The total time < 2½
• 25 2s + 20 s < 2½

So his average speed running is greater than 13 km/h and his average speed cycling is greater than 26 km/h

In this example we get to use two inequalities at once:

Example: The velocity v m/s of a ball thrown directly up in the air is given by v = 20 − 10t , where t is the time in seconds. At what times will the velocity be between 10 m/s and 15 m/s?

• velocity in m/s: v
• the time in seconds: t
• v = 20 − 10t

We are being asked for the time t when v is between 5 and 15 m/s:

So the velocity is between 10 m/s and 15 m/s between 0.5 and 1 second after.

And a reasonably hard example to finish with:

Example: A rectangular room fits at least 7 tables that each have 1 square meter of surface area. The perimeter of the room is 16 m. What could the width and length of the room be?

Make a sketch: we don't know the size of the tables, only their area, they may fit perfectly or not!

• the length of the room: L
• the width of the room: W

The formula for the perimeter is 2(W + L) , and we know it is 16 m

• 2(W + L) = 16
• L = 8 − W

We also know the area of a rectangle is the width times the length: Area = W × L

And the area must be greater than or equal to 7:

• W × L ≥ 7

We are being asked for the possible values of W and L

Let's solve:

So the width must be between 1 m and 7 m (inclusive) and the length is 8−width .

• Say W = 1, then L = 8−1 = 7, and A = 1 x 7 = 7 m 2 (fits exactly 7 tables)
• Say W = 0.9 (less than 1), then L = 7.1, and A = 0.9 x 7.1 = 6.39 m 2 (7 won't fit)
• Say W = 1.1 (just above 1), then L = 6.9, and A = 1.1 x 6.9 = 7.59 m 2 (7 fit easily)
• Likewise for W around 7 m

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Writing Inequalities from Word Problems

Learn about writing inequalities from word problems with help from our practice examples. If you want to test yourself, or get some practice, then try one of our graded worksheets, or our online quiz.

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What is an Inequality?

The language of inequalities.

• How to Write an Inequality from a Word Problem
• Write an Inequality from a Word Problems Examples

Writing Inequalities from Word Problems Worksheets

• Writing Inequalities from Word Problems Online Quiz
• More related resources

An inequality is when you have a relationship between two values of expressions which are not equal to each other.

There are a few different options for different types of inequalities:

• Greater than (>) where one expression or value is greater than another, e.g. 7 > 5
• Less than (<) where one expression or value is less than another, e.g. 9 < 2 x 6
• Greater than or equal to (≥) where one expression or value is greater than or equal to another, e.g. 20 + 4 ≥ 17
• Less than or equal to (≤) where one expression or value is less than or equal to another, e.g. 18 ≤ 9 x 2
• Not equal to (≠) where one expression or value is not equal to another, e.g. 7 ≠ 4

When writing inequalities from word problems, we have to look carefully at and understand the language being used.

Different words and phrases have different meanings when deciding on which inequality to use.

The mathematical notation is really just a shorthand way of writing the words more efficiently and clearly.

Here is a quick table showing some of the written expressions often used and which inequality they are represented by.

Note: the word 'between' is mainly used to mean between inclusively (including end points).

However, sometimes 'between' is used to mean between exclusively (excluding end points).

To avoid ambiguity, it is good practice to include the word 'inclusive' or 'exclusive' to make it completely clear if the end points are included or not.

Some simple examples showing inequalities from phrases:

The variable names (letters) have been chosen at random - you can use any variable name to represent any value.

Note: you need to read the word problem carefully because sometimes the inequality does not match the language used, especially when the inequality involves finding out what is left over or what remains after an amount is taken away. See Examples 2) and 7) below.

How to Write Inequalities from Word Problems

When we are writing an inequality from a word problem, we are basically translating the word problem into mathematical language and symbols.

When writing an inequality from a word problem, there are two simple steps you need to follow...

Step 1) Read the word problem carefully and change the word problem into algebra.

• use the language of inequalities table to help you select the right inequality

Step 2) Use algebra to solve the word problem

Step 3) rewrite the inequality using algebra., write an inequality from a word problem examples.

The best way to learn how to write inequalities from word problems and see how they work is to look at some ready made examples.

Writing Inequalities from Word Problems - Basic Examples

Here are some examples of writing inequalities from word problems.

Example 1) Sally bakes some cookies and needs to put them in the over for at least 12 minutes. Write an inequality using the variable t to show how long the cookies need to be baked in the oven.

The vocabulary which tells us about the inequality are the words: at least .

This means we need to use the ≥ symbol.

So the inequality is t ≥ 12 minutes

Example 2) Newton has a 30 ounce bottle of water. He drinks over half of the bottle. Write an inequality using the variable c to show how many ounces are left in the bottle.

The vocabulary which tells us about the inequality are the words: over .

However, because he has drunk over half the bottle, it means that there is under half a bottle left.

So the symbol we need is < and the amount is ½ of 30 = 15.

So the inequality is b < 15 ounces

Example 3) Anna is more than three times as old as Bertie. If Bertie is 8 years old, write an inequality using the variable A to show how old Anna is.

The vocabulary which tells us about the inequality are the words: more than .

So the symbol we need is > and the amount is 3 x 8 = 24.

So the inequality is A > 24 years old

Example 4) A book has 14 chapters.The shortest chapter has 12 pages. Write an inequality using the variable p to show how many pages the book has.

The vocabulary which tells us about the inequality are the words: shortest .

If the shortest chapter has 11 pages, then there must be some chapters with more than 11 pages.

So the symbol we need is > and the amount is 14 x 12 = 168.

So the inequality is p > 168 pages.

Writing Inequalities from Word Problems -Intermediate Examples

These examples use two different variables and express one variable in terms of another.

Example 5) Captain and Frazer have some gold coins. Captain has at least three times as many coins as Frazer. Write an inequality for the number of coins Captain has (c) in terms of the number of coins Frazer has (f).

So the symbol we need is ≥

So the inequality is c ≥ 3f.

Example 6) In a hotel there are f flights of stairs. Each flight has a maximum of 12 steps. There are also 3 steps up to the main entrance. Write an expression for the total number of steps, s, in terms of f.

The vocabulary which tells us about the inequality are the words: a maximum of .

So the symbol we need is ≤

We know that there are f flights of steps and also 3 extra steps.

So the inequality is s ≤ 12f + 3.

Writing Inequalities from Word Problems - Harder Examples

These examples involve solving word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers.

There are also examples where the variable lies between two values.

Example 7) Captain has a one-liter bottle of water. He drinks more than one-quarter of the bottle but less than one-half of the bottle. Write an inequality using the variable b to show the amount of water than is left in the bottle.

The vocabulary which tells us about the inequality are the words: more than and less than .

However, because we are looking at what is left in the bottle, rather than what has been drunk, we need to think carefully about the inequalities!

He drinks more than one-quarter of the bottle, so there will be less than three-quarters of the bottle left, so we need the symbol <

He drinks less than one-half of the bottle, so there will be one-half or more of the bottle left, so we need the symbol ≥

Half of the bottle = ½ liters = 500 ml. 1000 - 500 = 500 ml

Quarter of the bottle = ¼ liters = 250 ml. 1000 - 250 = 750 ml

So the inequality is b ≥ 500 ml and b < 750 ml This can be simplified to: 500 ≤ b < 750 ml This means that he has at least 500 ml but less than 750 ml left.

Example 8) Captain has challenged himself to catch a minimum of 50 fish from a lake. He manages to catch 8 of them and put them in his bucket. If he catches 6 fish every hour, write an inequality to show the time (t) in hours it will take him to reach his target.

The vocabulary which tells us about the inequality are the words: a minimum of .

The inequality we get from this problem is 6t + 8 ≥ 50

We are not finished yet, because this needs to be simplified and written in terms of t.

6t + 8 ≥ 50 so 6t ≥ 42

If we divide both sides of this inequality by 6, we get:

So the inequality is t ≥ 7 hours He needs to fish for at least 7 hours to reach his target.

Example 9) It takes Newton between 23 and 28 seconds (inclusive) to swim a length of a swimming pool. Write an inequality using the variable t to show how long it will take him to swim 3 lengths.

The vocabulary which tells us about the inequality are the words: between (inclusive) .

So the symbol we need is ≤ and ≥

3 x 23 = 69 and 3 x 28 = 84

So the inequality is t ≥ 69 and t ≤ 84 This can be simplified to: 69 ≤ t ≤ 84 It will take him between 69 and 84 seconds (inclusive) to swim 3 lengths.

We have a range of different inequality worksheets which involve writing inequalities from a range of word problems..

We have split the sheets into 3 sections: A, B and C

• Section A involves basic level questions aimed at 6th grade
• Section B involves medium level questions aimed at 6th and 7th grade
• Section C involves more advanced questions aimed at 7th and 8th grade

Writing Inequalities from Word Problems - Section A Easier

Sheet 1 involves picking the vocabulary and relevant information from the problem and writing the inequality

Sheet 2 involves the same skills as Sheet 1, but also involves an arithmetic operation to get the inequality.

• Inequalities from Word Problems Sheet A1
• PDF version
• Inequalities from Word Problems Sheet A2

Writing Inequalities from Word Problems - Section B Medium

Sheet 1 involves using two variables and writing an inequality for one variable in terms of the other variable

Sheet 2 is similar to Sheet 1 but with slightly harder problems.

• Inequalities from Word Problems Sheet B1
• Inequalities from Word Problems Sheet B2

Writing Inequalities from Word Problems - Section C

Sheet 1 involves using one variables and using the information to solve the inequality, usually in the form px + q > r or px + q < r, where p, q, and r are specific rational numbers

Sheet 2 involves the same skills as Sheet 1 but has compound inequalities in each question

• Inequalities from Word Problems Sheet C1
• Inequalities from Word Problems Sheet C2

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Inequalities on a Number Line Worksheets

• Inequalities on a Number Line

6th Grade Ratio and Unit Rate Worksheets

These 5th grade ratio worksheets are a great way to introduce this concept.

We have a range of part to part ratio worksheets and slightly harder problem solving worksheets.

• Ratio Part to Part Worksheets
• Ratio and Proportion Worksheets
• The Definition of Unit Rate
• Unit Rate Problems 6th Grade

6th Grade Algebra Worksheets

If you are looking for some 6th grade algebra worksheets to use with your child to help them understand simple equations then try our selection of basic algebra worksheets.

There are a range of 6th grade math worksheets covering the following concepts:

• Generate the algebra - and write your own algebraic expressions;
• Calculate the algebra - work out the value of different expressions;
• Solve the algebra - find the value of the term in the equation.
• Use the distributive property to factorize and expand different expressions
• 6th Grade Distributive Property Worksheets
• Expressions and Equations 6th Grade
• Basic Algebra Worksheets (6th & 7th Grade)

Writing Inequalities from Word Problems Quiz

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This quick quiz tests your skill at writing inequalities from a range of word problems.

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Solving Word Problems in Algebra Inequality Word Problems

How are you with solving word problems in Algebra? Are you ready to dive into the "real world" of inequalities? I know that solving word problems in Algebra is probably not your favorite, but there's no point in learning the skill if you don't apply it.

I promise to make this as easy as possible. Pay close attention to the key words given below, as this will help you to write the inequality. Once the inequality is written, you can solve the inequality using the skills you learned in our past lessons.

I've tried to provide you with examples that could pertain to your life and come in handy one day. Think about others ways you might use inequalities in real world problems. I'd love to hear about them if you do!

Before we look at the examples let's go over some of the rules and key words for solving word problems in Algebra (or any math class).

Word Problem Solving Strategies

• Read through the entire problem.
• Highlight the important information and key words that you need to solve the problem.
• Identify your variables.
• Write the equation or inequality.
• Write your answer in a complete sentence.
• Check or justify your answer.

I know it always helps too, if you have key words that help you to write the equation or inequality. Here are a few key words that we associate with inequalities! Keep these handy as a reference.

Inequality Key Words

• at least - means greater than or equal to
• no more than - means less than or equal to
• more than - means greater than
• less than - means less than

Ok... let's put it into action and look at our examples.

Example 1: Inequality Word Problems

Keith has $500 in a savings account at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws $25 each week for food, clothes, and movie tickets.

• Write an inequality that represents Keith's situation.
• How many weeks can Keith withdraw money from his account? Justify your answer.

Step 1: Highlight the important information in this problem.

Note:  At least is a key word that notes that this problem must be written as an inequality.

Step 2 : Identify your variable. What don't you know? The question verifies that you don't know how many weeks.

Let w = the number of weeks

Step 3: Write your inequality.

500 - 25w > 200

I know you are saying, "How did you get that inequality?"

I know the "at least" part is tricky. You would probably think that at least means less than.

But... he wants the amount in his account to be at least $200 which means $200 or greater. So, we must use the greater than or equal to symbol.

Step 4 : Solve the inequality.

The number of weeks that Keith can withdraw money from his account is 12 weeks or less.

Step 5: Justify (prove your answer mathematically).

I'm going to prove that the largest number of weeks is 12 by substituting 12 into the inequality for w. You could also substitute any number less than 12.

Since 200 is equal to 200, my answer is correct. Any more than 12 weeks and his account balance would be less than $200.  Any number of weeks less than 12 and his account would stay above $200.

That wasn't too bad, was it? Let's take a look at another example.

Example 2: More Inequality Word Problems

Yellow Cab Taxi charges a $1.75 flat rat e in addition to $0.65 per mile . Katie has no more than $10 to spend on a ride.

• Write an inequality that represents Katie's situation.
• How many miles can Katie travel without exceeding her budget? Justify your answer.

Note:  No more than are key words that note that this problem must be written as an inequality.

Step 2 : Identify your variable. What don't you know? The question verifies that you don't know the number of miles Katie can travel.

Let m = the number of miles

Step 3:  Write the inequality.

0.65m + 1.75 < 10

Are you thinking, "How did you write that inequality?"

The "no more than" can also be tricky. "No more than" means that you can't have more than something, so that means you must have less than!

Step 4: Solve the inequality.

Since this is a real world problem and taxi's usually charge by the mile, we can say that Katie can travel 12 miles or less before reaching her limit of $10.

Are you ready to try some on your own now? Yes... of course you are! Click here to move onto the word problem practice problems.

Take a look at the questions that other students have submitted:

Quite a complicated problem about perimeter and area of a rectangle

This is a toughie.... a compound inequalities word problem

• Inequalities
• Inequality Word Problems

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Calcworkshop

Solving Inequality Word Problems with 5 Awesome Examples!

// Last Updated: January 20, 2020 - Watch Video //

Throughout this lesson we are going to become masters at being able to write a Linear Inequality given a real life situation, and interpret out results.

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We begin this lesson with a quick review of our three basic steps for solving any type of word problem:

• Read the problem carefully, noting punctuation, and determine the question being asked.
• Create a sidebar, by identifying all important information and translating words into algebraic expressions .
• Form one Inequality statement and solve.

Example of How to writing an Inequality

By using these three surefire steps, we will quickly see that Inequality Word Problems are more than manageable.

In fact, we will learn how to handle such problems dealing with multiple unknowns, evaluate expressions, and interpret our answers.

We will remind ourselves of our inequality key phrases, as Algebra Class so nicely summarizes, draw upon our knowledge of how to simplify expressions and solve inequalities using our SCAM technique.

Together we will look at five classic questions in detail, where we will practice writing linear inequalities with variables in order to describe real-word situations.

Inequality Word Problems (How-To) – Video

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Solving Inequalities Word Problems - Examples & Practice - Expii

Solving inequalities word problems - examples & practice, explanations (3), (videos) solving word problems involving inequalities.

by PatrickJMT

This video by Patrick JMT works through an inequality word problem. There is another video with a second example later in the explanation.

The first example is:

A widget factory has a fixed operating cost of $3,600 per day plus costs $1.40 per widget produced. If a widget sells for $4.20, what is the least number of widgets that must be sold per day to make a profit?

To make a profit, the revenue received needs to be greater than the cost needed to make it. \text{Cost}">Revenue>Cost

Now, we want to create an expression for each of these. Let's let x be the number of widgets produced.

For cost, we know there's a fixed cost of $3600 and then $1.40 for each widget. 3600+1.4x">Revenue>3600+1.4x

For revenue, we receive $4.20 for each widget. 3600+1.4x">4.2x>3600+1.4x

Finally, we solve by getting x to one side of the inequality, similar to how you'd do for an equation . 3600+1.4x\\ 4.2\color{#8925AE}{-1.4x}&>3600+1.4x\color{#8925AE}{-1.4x}\\ 2.8x&>3600\\ x&>1285.71 \end{align}">4.2x>3600+1.4x4.2−1.4x>3600+1.4x−1.4x2.8x>3600x>1285.71

So, in order to make a profit, you'd need to sell over 1285 widgets.

Here's another example.

Jason makes and sells fishing poles. If he has costs totaling $12,000 per year plus a cost of $4 per pole, how many fishing poles must he produce to make a profit of $48,000 in one year, if he sells the fishing poles for $28 each?

We want the profit to be greater than or equal to $48,000. Profit≥48000

We can also rewrite profit as the amount you make minus the amount you spend. amount made−amount spent≥48000

If he sells the poles for $28 each, then that is the amount he makes. We can let x be the number of poles. 28x−amount spent≥48000

We also know there's a fixed cost to make the poles for $12000. Also, each pole costs $4 to make. 28x−(12000+4x)≥48000

Finally, we can solve the inequality by getting x alone on one side. 28x−(12000+4x)≥4800028x−12000−4x≥4800024x−12000≥4800024x−12000+12000≥48000+1200024x≥60000x≥2500

To get here, you may want to review the distributive properties with inequalities and solving inequalities via grouping .

Jason will need to sell at least 2,500 poles in order to make a profit of $48,000.

Related Lessons

Inequality word problems.

The key to word problems is translating the given information into math. In this case, we need to translate word problems into statements of inequality .

Inequalities are denoted by words like "greater than" and "less than". In general, a comparison is made in problems with inequalities.

Let's look at two examples:

Image source: by Hannah Bonville

We need to convert this information into an inequality. We are trying to figure out how many ducks Chris can by while staying at or under budget. In other words, we want to keep the cost less than or equal to his budget. Sounds like a good place to use a ≤ sign, doesn't it?

What inequality translates the information in this word problem? Let x be the number of ducks Chris can buy.

6.95x+9.99≤80

6.95+9.99x≤80

Solving Word Problems Involving Inequalities

Inequality word problems will always involve comparisons . In real life, you might use inequalities to decide which of two options is better. Look for comparison words to figure out which kind of symbol you should use. Look for rates to write out an equation .

Test Yourself: What are the comparison words in the following problem, and what inequality symbol do they relate to?

Billie earns a salary of $18,000 per year plus a 5% commission on all her sales. How much must her sales be if her annual income is to be no less than $20,000?

"Plus" ">>

"No less than" ≥

"No less than" ≤

"How much" <

Free Linear Inequalities Word Problems Worksheet

While teaching high school, one of the biggest struggles I have seen students face is applying math concepts to the real-world. This is true when it comes to solving linear inequalities word problems too. Students often find it challenging to grasp how these mathematical principles translate into practical, everyday life scenarios.

That’s why I have put together this linear inequalities word problems worksheet! My goal is to help you learn a few tips and tricks and practice applying linear inequalities to the real-world! 

What are Linear Inequality Word Problems?

Lesson plans that focus on linear inequality word problems typically show students how to apply the skills they developed while solving inequalities to the real-world. There are a wide variety of inequality applications, ranging from social studies to physical science. Regardless of the application, the idea is that you will be faced with a word problem that requires you to model the scenario using a linear inequality.

In general, linear inequality word problems describe how one quantity has to be less than or greater than another. Your goal is then to use inequality symbols and  algebraic expressions  to represent the scenario algebraically.

For most, the concept of a linear inequality is first introduced in middle school (although this will vary by curriculum). My daughter, for example, is in 2nd grade math and is just starting to explore these problems. Others may not see this concept until later in high school (in some cases not until the 12th grade).

How to Solve a Linear Inequality Word Problem

The best way to solve any math word problem is to start by reading the question very carefully, and linear inequality word problems are no different!

I always encourage my students to underline or highlight any key words and important information. When it comes to how to solve a linear inequality word problem, the key words usually help you understand:

• which quantities you are working with
• whether you are working with  less than  or  greater than symbols

Once you have identified this important information, your goal is to write an algebraic expression using an inequality symbol that models the scenario. You can then solve the inequality using a similar process to what you would apply when solving one-step equations or  two step equations .

There are many different ways to represent the solution to an inequality problem. Sometimes you will be asked to use a number line, which shows all the negative values or positive values that belong to a solution set. The worksheet attached below will provide you with some practice using number lines to communicate your answers to inequality word problems.

​Solving a Linear Inequality Word Problem Example

Age problems are common applications that you will see when solving linear inequality word problems. These types of problems can be simple or complex, but I wanted to start by sharing a simple one here so that you can understand the basics of how to solve linear inequality word problems.

A father is 3 times as old as his son, but three times his son’s age is less than 30. What is the oldest the son can be?

We can begin by calling out the key words that give us important information. In this case, the following two pieces of information are considered important to the problem:

• “3 times as old as”
• “less than 30”

This tells us that we will be working with a “<” symbol, and multiplication of a quantity by 3. If we let  n  represent the age of the son, we can set up a linear inequality as follows:

$$3n<30$$

Reading this statement in English tells us that “3 times the son’s age is less than 30”. If you head back to the original problem, that seems to match the scenario given, doesn’t it? Great! That tells us that we have a good algebraic model for our real-world problem! Let’s move on and start solving!

Remember that we can solve a linear inequality using algebra in a similar way to solving a linear equation. This means that we can add or subtract terms on both sides of the inequality symbol, and we can also multiply and divide terms on both sides of the inequality symbol.

In this case, since we are multiplying  n  by 3, we divide both sides by 3 to isolate  n.

$$\frac{3n}{3}< \frac{30}{3}$$

$$n<10$$

This tells us that the son’s age must be less than 10 in order for the father to be three times his age but still less than 30 years old himself.

We can test this by multiplying a number greater than 10 by 3. For example, \(11 \times 3 = 33\). Notice that 33 is  not  less than 30. Therefore the son cannot be 11. The only values that will make this inequality statement true are values that are less than (not including) 10.

We can represent this solution on a number line by placing a hollow circle at 10 and drawing an arrow to the left toward the negative values. However, we should stop at zero since the son’s age cannot be less than zero. Note that the son must also not be equal to zero. If he were, the father would be \(3 \times 0 = 0\) as well!

If you need more practice with the algebra strategies that can be used to solve inequalities, check out this collection of  solving linear inequalities worksheets .

Linear Inequalities Word Problems Worksheet

Now that you have had some practice applying your understanding of linear inequalities to solve a real-world problem, you are ready to practice! Below I have included a linear inequalities word problem worksheet that covers a variety of problem types, ranging from age problems, to a bake sale problem with pink cupcakes!

As promised, this worksheet will also provide you with practice representing the solution to a linear inequality word problem on a number line. While you should be sure to attempt every problem as independent work first, make sure that you also check the answer key! This is an important step to make sure that you fully understand each problem.

My hope is that you find this linear inequalities word problems worksheet helpful as independent work whether you are in 1st grade math, 7th grade math, or anything in between!

Download the PDF worksheet by clicking below!

Practice Solving Linear Inequalities Word Problems

I hope this linear inequalities word problems worksheet has provided you with some practice with applying this important math concept to the real-world! My goal was to share examples that cover a variety of areas of life. Hopefully solving these problems also gave you an appreciation for how linear inequalities can be used in everyday life.

Whether you first learned about linear inequality word problems in the 6th grade or are just experiencing them for the first time in the 12th grade, the most important thing you can do is practice. Solving many different types of linear inequalities word problems will help you start to recognize patterns. This will help you with the first initial step of writing the inequality as an algebraic statement, something many students find challenging! 

With consistent practice, you’ll develop a strong foundation in solving linear inequalities, enhancing your problem-solving skills and confidence in applying mathematical concepts to diverse real-world scenarios. Keep exploring and practicing, and you’ll find that handling linear inequalities becomes more intuitive over time!

If you are looking for more word problem resources, check out this linear equations word problems worksheet !

Did you find this linear inequalities word problems worksheet helpful? Share this post and subscribe to Math By The Pixel on YouTube for more helpful mathematics content!

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Examples of Solving Harder Linear Inequalities

Intro & Formatting Worked Examples Harder Examples & Word Prob's

Once you'd learned how to solve one-variable linear equations, you were then given word problems. To solve these problems, you'd have to figure out a linear equation that modelled the situation, and then you'd have to solve that equation to find the answer to the word problem.

Content Continues Below

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Solving Inequalities

So, now that you know how to solve linear inequalities — you guessed it! — they give you word problems.

• The velocity of an object fired directly upward is given by V = 80 − 32 t , where the time t is measured in seconds. When will the velocity be between 32 and 64 feet per second (inclusive)?

This question is asking when the velocity, V , will be between two given values. So I'll take the expression for the velocity,, and put it between the two values they've given me. They've specified that the interval of velocities is inclusive, which means that the interval endpoints are included. Mathematically, this means that the inequality for this model will be an "or equal to" inequality. Because the solution is a bracket (that is, the solution is within an interval), I'll need to set up a three-part (that is, a compound) inequality.

I will set up the compound inequality, and then solve for the range of times t :

32 ≤ 80 − 32 t ≤ 64

32 − 80 ≤ 80 − 80 − 32 t ≤ 64 − 80

−48 ≤ −32 t ≤ −16

−48 / −32 ≥ −32 t / −32 ≥ −16 / −32

1.5 ≥ t ≥ 0.5

Note that, since I had to divide through by a negative, I had to flip the inequality signs.

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Note also that you might (as I do) find the above answer to be more easily understood if it's written the other way around, with "less than" inequalities.

And, because this is a (sort of) real world problem, my working should show the fractions, but my answer should probably be converted to decimal form, because it's more natural to say "one and a half seconds" than it is to say "three-halves seconds". So I convert the last line above to the following:

0.5 ≤ t ≤ 1.5

Looking back at the original question, it did not ask for the value of the variable " t ", but asked for the times when the velocity was between certain values. So the actual answer is:

The velocity will be between 32 and 64 feet per second between 0.5 seconds after launch and 1.5 seconds after launch.

Okay; my answer above was *extremely* verbose and "complete"; you don't likely need to be so extreme. You can probably safely get away with saying something simpler like, "between 0.5 seconds and 1.5 seconds". Just make sure that you do indeed include the approprioate units (in this case, "seconds").

Always remember when doing word problems, that, once you've found the value for the variable, you need to go back and re-read the problem to make sure that you're answering the actual question. The inequality 0.5 ≤  t  ≤ 1.5 did not answer the actual question regarding time. I had to interpret the inequality and express the values in terms of the original question.

• Solve 5 x + 7 < 3( x  + 1) .

First I'll multiply through on the right-hand side, and then solve as usual:

5 x + 7 < 3( x + 1)

5 x + 7 < 3 x + 3

2 x + 7 < 3

2 x < −4

x < −2

In solving this inequality, I divided through by a positive 2 to get the final answer; as a result (that is, because I did *not* divide through by a minus), I didn't have to flip the inequality sign.

• You want to invest $30,000 . Part of this will be invested in a stable 5% -simple-interest account. The remainder will be "invested" in your father's business, and he says that he'll pay you back with 7% interest. Your father knows that you're making these investments in order to pay your child's college tuition with the interest income. What is the least you can "invest" with your father, and still (assuming he really pays you back) get at least $1900 in interest?

First, I have to set up equations for this. The interest formula for simple interest is I = Prt , where I is the interest, P is the beginning principal, r is the interest rate expressed as a decimal, and t is the time in years.

Since no time-frame is specified for this problem, I'll assume that t  = 1 ; that is, I'll assume (hope) that he's promising to pay me at the end of one year. I'll let x be the amount that I'm going to "invest" with my father. Then the rest of my money, being however much is left after whatever I give to him, will be represented by "the total, less what I've already given him", so 30000 −  x will be left to invest in the safe account.

Then the interest on the business investment, assuming that I get paid back, will be:

I = ( x )(0.07)(1) = 0.07 x

The interest on the safe investment will be:

(30 000 − x )(0.05)(1) = 1500 − 0.05 x

The total interest is the sum of what is earned on each of the two separate investments, so my expression for the total interest is:

0.07 x + (1500 − 0.05 x ) = 0.02 x + 1500

I need to get at least $1900 ; that is, the sum of the two investments' interest must be greater than, or at least equal to, $1,900 . This allows me to create my inequality:

0.02 x + 1500 ≥ 1900

0.02 x ≥ 400

x ≥ 20 000

That is, I will need to "invest" at least $20,000 with my father in order to get $1,900 in interest income. Since I want to give him as little money as possible, I will give him the minimum amount:

I will invest $20,000 at 7% .

• An alloy needs to contain between 46% copper and 50% copper. Find the least and greatest amounts of a 60% copper alloy that should be mixed with a 40% copper alloy in order to end up with thirty pounds of an alloy containing an allowable percentage of copper.

This is similar to a mixture word problem , except that this will involve inequality symbols rather than "equals" signs. I'll set it up the same way, though, starting with picking a variable for the unknown that I'm seeking. I will use x to stand for the pounds of 60% copper alloy that I need to use. Then 30 −  x will be the number of pounds, out of total of thirty pounds needed, that will come from the 40% alloy.

Of course, I'll remember to convert the percentages to decimal form for doing the algebra.

How did I get those values in the bottom right-hand box? I multiplied the total number of pounds in the mixture ( 30 ) by the minimum and maximum percentages ( 46% and 50% , respectively). That is, I multiplied across the bottom row, just as I did in the " 60% alloy" row and the " 40% alloy" row, to get the right-hand column's value.

The total amount of copper in the mixture will be the sum of the copper contributed by each of the two alloys that are being put into the mixture. So I'll add the expressions for the amount of copper from each of the alloys, and place the expression for the total amount of copper in the mixture as being between the minimum and the maximum allowable amounts of copper:

13.8 ≤ 0.6 x + (12 − 0.4 x ) ≤ 15

13.8 ≤ 0.2 x + 12 ≤ 15

1.8 ≤ 0.2 x ≤ 3

9 ≤ x ≤ 15

Checking back to my set-up, I see that I chose my variable to stand for the number of pounds that I need to use of the 60% copper alloy. And they'd only asked me for this amount, so I can ignore the other alloy in my answer.

I will need to use between 9 and 15 pounds of the 60% alloy.

Per yoozh, I'm verbose in my answer. You can answer simply as " between 9 and 15 pounds ".

• Solve 3( x − 2) + 4 ≥ 2(2 x − 3)

First I'll multiply through and simplify; then I'll solve:

3( x − 2) + 4 ≥ 2(2 x − 3)

3 x − 6 + 4 ≥ 4 x − 6

3 x − 2 ≥ 4 x − 6

−2 ≥ x − 6            (*)

Why did I move the 3 x over to the right-hand side (to get to the line marked with a star), instead of moving the 4 x to the left-hand side? Because by moving the smaller term, I was able to avoid having a negative coefficient on the variable, and therefore I was able to avoid having to remember to flip the inequality when I divided through by that negative coefficient. I find it simpler to work this way; I make fewer errors. But it's just a matter of taste; you do what works for you.

Why did I switch the inequality in the last line and put the variable on the left? Because I'm more comfortable with inequalities when the answers are formatted this way. Again, it's only a matter of taste. The form of the answer in the previous line, 4 ≥ x , is perfectly acceptable.

As long as you remember to flip the inequality sign when you multiply or divide through by a negative, you shouldn't have any trouble with solving linear inequalities.

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How to Solve Word Problems Involving One-step Inequalities?

In this step-by-step guide, you will learn how to solve word problems involving one-step inequalities.

A Step-by-step Guide to Solve Word Problems Involving One-step Inequalities

A word problem involving one-step inequalities is a type of mathematical problem that describes a situation using words and requires solving an inequality that can be solved with just one step.

In other words, the problem can be solved by performing one mathematical operation, such as adding, subtracting, multiplying, or dividing.

One-step inequalities involve comparing two values using one of the inequality symbols, such as \(<\) (less than), \(>\) (greater than), \(≤\) (less than or equal to), or \(≥\) (greater than or equal to).

The inequality symbol is used to represent the relationship between the two values in the problem.

Here’s a step-by-step guide to solving word problems involving one-step inequalities:

Step 1: Understand the problem

Read the problem carefully and try to understand what it is asking you to do. Identify the unknown quantity and determine what it is that you are being asked to find.

Make note of any inequality symbols \((>, <, ≥, ≤)\) that are present in the problem.

Step 2: Translate the problem into an inequality

Once you have understood the problem, translate it into an inequality. Use the inequality symbol that matches the problem’s direction.

For example, if the problem says “less than”, use \(<\).

If it says “greater than”, use \(>\).

Step 3: Solve the inequality

Solve the inequality by performing the same operation on both sides of the inequality.

Remember that if you multiply or divide by a negative number, you must reverse the inequality symbol.

If you add or subtract a negative number, it is the same as adding or subtracting a positive number.

Step 4: Check your solution

Check your solution by plugging it back into the original inequality.

If the inequality is true, your solution is correct. If not, recheck your work and try again.

Step 5: Write the answer in a sentence

Write the solution to the problem in a complete sentence, using the correct units of measurement if applicable.

For example, let’s say you are given the problem: “Sam has at least \($20\) in his pocket. Write an inequality to represent this situation.”

The unknown quantity is the amount of money Sam has in his pocket. We need to write an inequality to represent the situation.

Sam has at least \($20\) in his pocket, so the inequality can be written as: \(x\ge 20\)

There is no need to solve inequality since it is already in its simplest form.

If we plug in any value greater than or equal to \(20\), the inequality holds. For example, if Sam has \($25\) in his pocket, the inequality \(25 ≥ 20\) is true.

Sam has at least \($20\) in his pocket.

Solving Word Problems Involving One-step Inequalities – Examples 1

An athlete runs at least \(4\) miles per day to train for a marathon. If the athlete has already run \(2\) miles today, how many more miles does the athlete need to run to meet the daily training goal?

To solve this problem, we can set up an inequality:

Miles run so far \(+\) Additional miles run \(≥\) Daily training goal

\(2 + x ≥ 4\)

We subtract \(2\) from both sides to isolate the variable:

Therefore, the athlete needs to run at least \(2\) more miles to meet the daily training goal.

Solving Word Problems Involving One-step Inequalities – Examples 2

“Maria needs to save at least \($50\) to buy a new pair of shoes. She has already saved \($30\). How much more money does she need to save?”

Money saved so far \(+\) Additional money saved \(≥\) Total money needed

\($30 + x ≥ $50\)

We subtract \($30\) from both sides to isolate the variable:

\(x ≥ $20\)

Therefore, Maria needs to save at least \($20\) more to be able to buy the new pair of shoes.

by: Effortless Math Team about 1 year ago (category: Articles )

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Inequalities Word Problems Worksheets

Inequalities word problems worksheets can help encourage students to read and think about the questions, rather than simply recognizing a pattern to the solutions.Inequalities word problems worksheet come with the answer key and detailed solutions which the students can refer to anytime.

Benefits of Inequalities Word Problems Worksheets

Inequalities word problems worksheets help kids to improve their speed, accuracy, logical and reasoning skills.

Inequalities word problems worksheets gives students the opportunity to solve a wide variety of problems helping them to build a robust mathematical foundation. Inequalities word problems worksheets helps kids to improve their speed, accuracy, logical and reasoning skills in performing simple calculations related to the topic of inequalities.

Inequalities word problems worksheets are also helpful for students to prepare for various competitive exams.

These worksheets come with visual simulation for students to see the problems in action, and provides a detailed step-by-step solution for students to understand the process better, and a worksheet properly explained about the Inequalities.

Download Inequalities Word Problems Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

☛ Check Grade wise Inequalities Word Problems Worksheets

• 7th Grade Inequalities Worksheets

SOLVING WORD PROBLEMS WITH INEQUALITIES

Many of the real-life word problems can be solved algebraically by translating the given information into an inequality and then solving the inequality.

Example 1 :

Alex Car Rental charges a flat fee of $40.00 per day plus $0.54 per mile to rent a car. Jack Car Rental charges a flat fee of $65.00 per day plus $0.36 per mile to rent a car. If a car is rented for three days, at least how many miles would you have to drive, to the nearest mile, to make the Jack Car Rental company the better option ?

The problem asks for how many miles you would have to drive to make Jack Car Rental the better option.

Let x be the number of miles of driving which would make Jack Car Rental the better option. 

Alex Car Rental's rental charge for 3 days 

Jack Car Rental's rental charge for 3 days 

40  ⋅ 3 + 0.54x  >  65 ⋅ 3 + 0.36x

Simplify. 

120 + 0.54x  >  195 + 0.36x

Subtract 0.36x from each side. 

120 + 0.18x  >  195

Subtract 120 from each side. 

0.18x  >  75

Divide each side by 0.18.

x  >  416.67

You need to drive 417 miles or more to make Jack Car Rental the better option.

Example 2 :

A 38 inch long wire is cut into two pieces. The longer piece has to be at least 3 inches longer than twice the shorter piece. What is the maximum length of the shorter piece, to the nearest inch ?

The problem asks for the maximum length of the shorter piece, to the nearest inch.

Let x be the length of the shorter piece. 

Then, the length of the longer piece is (38 - x).

The longer piece is at least 3 inches longer than twice the shorter piece.

Then, 

38 - x   ≥  2x + 3

Add x to each side. 

38   ≥  3x + 3

Subtract 3 from each side. 

35   ≥  3x

Divide each side by 3. 

11.67   ≥  x

The maximum length of the shorter piece, to the nearest inch, is 11.

Example 3 :

Tom wants to rent a truck for two days and pay no more than $300. Find the maximum distance (in miles) Tom can drive the truck if the truck rental cost $49 a day plus $0.40 a mile.

Let x be the total number of miles travelled in 2 days and y be the total cost.

y = 0.40x + 2(49)

y = 0.40x + 98

Given : The total cost should be no more than $300.

y  ≤ 300

0.40x + 98  ≤ 300

0.4x  ≤ 202

x ≤ 505

The maximum distance Tom can drive is 505 miles.

Example 4 :

Tim has 140 paperback and hard cover copies in his book shelf. If the hard cover copies do not exceed one sixth the number of paperback copies, find the minimum number of paperback copies in Tim’s book shelf.

Let p  be the number of paper back copies and h be the number hard cover copies.

Given : Tim has 140 paperback and hard cover copies in his book shelf.

p + h = 140

h = 140 - p

Given : The hard cover copies do not exceed one sixth the number of paperback copies.

h  ≤  ( ⅙ ) p

h  ≤   ᵖ⁄₆

p ≥ 6h

Substitute  h = 140 - p.

p ≥ 6(140 - p)

p ≥ 840 - 6p

7p ≥ 840

p ≥ 120

The minimum number of paperback copies in Tim’s book shelf is 120.

Example 5 :

The number of students in a geometry class is four fifths the number of students in a Spanish class. The total number of students in both classes does not exceed 54. What is the greatest possible number of students in the Spanish class?

Let g  be the number of students in geometry class and s be the number of students in Spanish class.

Given : The number of students in geometry class is four fifths the number of students in Spanish class.

Given : The total number of students in both classes does not exceed 54.

g + s  ≤ 54

Multiply both sides by 5.

5(g + s)  ≤ 5(54)

5g + 5s ≤ 270

Substitute 5g = 4s.

4s + 5s ≤ 270

The greatest possible number of students in the Spanish class is 30.

Example 6 :

At a sporting goods store, Jay paid $172 for a pair of shoes and a pair of pants. The pants cost less than two thirds of what the shoes cost. What is the minimum price of the shoes to the nearest dollar?

Let s be the price of shoes and p be the price of pants.

Given : Jay paid $172 for a pair of shoes and a pair of pants.

s + p = 172

p = 172 - s

Given :  The pants cost less than two thirds of what the shoes cost.

p < ( ⅔ )s

Substitute p = 172 - s.

3(172 - s) < 2s

516 - 3s < 2s

516 < 5s

103.2 < s

s > 103.2

The minimum price of the shoes to the nearest dollar is $104.

Example 7 :

Ty earns $14 an hour working on weekdays and $21 an hour working on weekends. If he wants to make at least $600 by working a total of 36 hours in a week, to the nearest hour, at least how many hours does he need to work on the weekends?

x ---> number of hours Ty works on weekdays

y ---> number of hours Ty works on weekends

Given : Ty works a total of 36 hours in a week.

Given : Ty  wants to make at least $600.

14x + 21y ≥ 600

Substitute x = 36 - y.

14(36 - y) + 21y ≥ 600

504 - 14y + 21y ≥ 600

504 + 7y ≥ 600

7y ≥ 96

y ≥ ⁹⁶⁄₇  ≈ 13.74

Ty needs to work at least 14 hours on the weekends.

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Chapter 1: Sets

• Representation of a Set
• Types Of Sets
• Universal Sets
• Venn Diagram
• Operations on Sets
• Union of Sets

Chapter 2: Relations & Functions

• Cartesian Product of Sets
• Relation and Function
• Introduction to Domain and Range
• Piecewise Function
• Range of a Function

Chapter 3: Trigonometric Functions

• Measuring Angles
• Trigonometric Functions
• Trigonometric Functions of Sum and Difference of Two Angles

Chapter 4: Principle of Mathematical Induction

• Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

• Complex Numbers
• Algebra of Real Functions
• Algebraic Operations on Complex Numbers | Class 11 Maths
• Polar Representation of Complex Numbers
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Chapter 6: Linear Inequalities

• Compound Inequalities
• Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation - Linear Inequalities | Class 11 Maths
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Solving Linear Inequalities Word Problems

Chapter 7: permutations and combinations.

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Chapter 8: Binomial Theorem

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Chapter 9: Sequences and Series

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We are well versed with equations in multiple variables. Linear Equations represent a point in a single dimension, a line in a two-dimensional, and a plane in a three-dimensional world. Solutions to linear inequalities represent a region of the Cartesian plane. It becomes essential for us to know how to translate real-life problems into linear inequalities. 

Linear Inequalities 

Before defining the linear inequalities formally, let’s see them through a real-life situation and observe why their need arises in the first place. Let’s say Albert went to buy some novels for himself at the book fair. He has a total of Rs 200 with him. The book fair has a special sale policy which offers any book at Rs 70. Now he knows that he may not be able to spend the full amount on the books. Let’s say x is the number of books he bought. This situation can be represented mathematically by the following equation, 

70x < 200

Since he can’t spend all the amount on books, and also the amount spent by him will always be less than Rs 200. The present situation can only be represented by the equation given above. Now let’s study the linear inequalities with a formal description, 

Two real numbers or two algebraic expressions which are related by symbols such as ‘>’, ‘<‘, ‘≥’ and’≤’ form the inequalities. Linear inequalities are formed by linear equations which are connected with these symbols. These inequalities can be further classified into two parts:  Strict Inequalities: The inequalities with the symbols such as ‘>’ or ‘<‘. Slack Inequalities: The inequalities with the symbols such as ‘≥’ or ‘≤’.

Rules of Solving Linear Inequalities:

There are certain rules which we should keep in mind while solving linear inequalities. 

• Equal numbers can be added or subtracted from both sides of the inequality without affecting its sign.
• Both sides of Inequality can be divided or multiplied by any positive number but when they are multiplied or divided by a negative number, the sign of the linear inequality is reversed.

Now with this brief introduction to linear inequalities, let’s see some word problems on this concept. 

Sample Problems

Question 1: Considering the problem given in the beginning. Albert went to buy some novels for himself at the book fair. He has a total of Rs 200 with him. The book fair has a special sale policy which offers any book at Rs 70. Now he knows that he may not be able to spend the full amount on the books. Let’s say x is the number of books he bought. Represent this situation mathematically and graphically. 

Solution: 

We know that Albert cannot buy books for all the money he has. So, let’s say the number of books he buys is “x”. Then,  70x < 200 ⇒ x <  To plot the graph of this inequality, put x = 0.  0 <   Thus, x = 0 satisfies the inequality. So, the graph for the following inequality will look like, 

Question 2: Consider the performance of the strikers of the football club Real Madrid in the last 3 matches. Ronaldo and Benzema together scored less than 9 goals in the last three matches. It is also known that Ronaldo scored three more goals than Benzema. What can be the possible number of goals Ronaldo scored? 

Let’s say the number of goals scored by Benzema and Ronaldo are y and x respectively.  x = y + 3 …..(1) x + y < 9  …..(2)  Substituting the value of x from equation (1) in equation (2).  y + 3 + y < 9  ⇒2y  < 6  ⇒y < 3 Possible values of y: 0,1,2  Possible values of x: 3,4,5

Question 3: A classroom can fit at least 9 tables with an area of a one-meter square. We know that the perimeter of the classroom is 12m. Find the bounds on the length and breadth of the classroom. 

It can fit 9 tables, that means the area of the classroom is atleast 9m 2 . Let’s say the length of the classroom is x and breadth is y meters.  2(x + y) = 12 {Perimeter of the classroom} ⇒ x + y = 6  Area of the rectangle is given by,  xy > 9  ⇒x(6 – x) > 9  ⇒6x – x 2 > 9  ⇒ 0 > x 2 – 6x + 9  ⇒ 0 > (x – 3) 2 ⇒ 0 > x – 3  ⇒ x < 3  Thus, length of the classroom must be less than 3 m.  So, then the breadth of the classroom will be greater than 3 m. 

Question 4: Formulate the linear inequality for the following situation and plot its graph. 

Let’s say Aman and Akhil went to a stationery shop. Aman bought 3 notebooks and Akhil bought 4 books. Let’s say cost of each notebook was “x” and each book was “y”. The total expenditure was less than Rs 500. 

Cost of each notebook was “x” and for each book, it was “y”. Then the inequality can be described as, 3x + 4y < 500 Putting (x,y) → (0,0)  3(0) + 4(0) < 500 Origin satisfies the inequality. Thus, the graph of its solutions will look like, x.

Question 5: Formulate the linear inequality for the following situation and plot its graph. 

A music store sells its guitars at five times their cost price. Find the shopkeeper’s minimum cost price if his profit is more than Rs 3000. 

Let’s say the selling price of the guitar is y, and the cost price is x. y – x > 3000 ….(1) It is also given that,  y = 5x ….(2)  Substituting the value of y from equation (2) to equation (1).  5x – x > 3000  ⇒ 4x > 3000  ⇒  x >  ⇒ x> 750  Thus, the cost price must be greater than Rs 750. 

Question 6: The length of the rectangle is 4 times its breadth. The perimeter of the rectangle is less than 20. Formulate a linear inequality in two variables for the given situation, plot its graph and calculate the bounds for both length and breadth. 

Let’s say the length is “x” and breadth is “y”.  Perimeter = 2(x + y) < 20 ….(1) ⇒ x + y < 10  Given : x = 4y  Substituting the value of x in equation (1).  x + y < 10 ⇒ 5y < 10  ⇒ y < 2 So, x < 8 and y < 2. 

Question 7: Rahul and Rinkesh play in the same football team. In the previous game, Rahul scored 2 more goals than Rinkesh, but together they scored less than 8 goals. Solve the linear inequality and plot this on a graph.

The equations obtained from the given information in the question, Suppose Rahul scored x Number of goals and Rinkesh scored y number of goals, Equations obtained will be, x = y+2 ⇢ (1) x+y< 8 ⇢ (2) Solving both the equations, y+2 + y < 8 2y < 6 y< 3 Putting this value in equation (2), x< 5

Question 8: In a class of 100 students, there are more girls than boys, Can it be concluded that how many girls would be there?

Let’s suppose that B is denoted for boys and G is denoted for girls. Now, since Girls present in the class are more than boys, it can be written in equation form as, G > B  The total number of students present in class is 100 (given), It can be written as, G+ B= 100 B = 100- G Substitute G> B in the equation formed, G> 100 – G 2G > 100 G> 50 Hence, it is fixed that the number of girls has to be more than 50 in class, it can be 60, 65, etc. Basically any number greater than 50 and less than 100.

Question 9: In the previous question, is it possible for the number of girls to be exactly 50 or exactly 100? If No, then why?

No, It is not possible for the Number of girls to be exactly 50 since while solving, it was obtained that, G> 50  In any case if G= 50 is a possibility, from equation G+ B= 100, B = 50 will be obtained. This simply means that the number of boys is equal to the number of girls which contradicts to what is given in the question. No, it is not possible for G to be exactly 100 as well, as this proves that there are 0 boys in the class.

Question 10: Solve the linear inequality and plot the graph for the same,

7x+ 8y < 30

The linear inequality is given as, 7x+ 8y< 30  At x= 0, y= 30/8= 3.75 At y= 0, x= 30/7= 4.28 These values are the intercepts. The graph for the above shall look like,  Putting x= y/2, that is, y= 2x in the linear inequality, 7x + 16x < 30 x = 1.304 y = 2.609

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• \mathrm{Lauren's\:age\:is\:half\:of\:Joe's\:age.\:Emma\:is\:four\:years\:older\:than\:Joe.\:The\:sum\:of\:Lauren,\:Emma,\:and\:Joe's\:age\:is\:54.\:How\:old\:is\:Joe?}
• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
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• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

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