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Absolute value word problems

You may need to review the lesson about how to solve absolute value equations and absolute value inequalities .

Problem #1:

Your have money in your wallet, but you don't know the exact amount. When a friend asks you, you say that you have 50 dollars give or take 15. Use an absolute value equation to find least and biggest amount of money in your pocket? 

Let x be the possible amount of money in your pocket. 

|x - 50| =  15

Equation #1

x - 50 = 15

x - 50 + 50 = 15 + 50

Equation #2

x - 50 = -15

x - 50 + 50 = -15 + 50

The least amount is 35 dollars and the biggest amount is 65 dollars.

Problem #2:

The ideal diameter of a piece of metal rod is 2.50 inches with an allowable error of at most 0.05 inch. Which rod(s) will you pick?

A. 2.5167  inches   B.    2.4417 inches      C.  2.484  inches     D.  2.558 inches     

        Tricky absolute value word problems

Problem #3:

The ideal selling price of a Toyota is 25000. The dealer allows this price to vary 5%. What is the lowest price this dealer can sell this Toyota??

The little trick is to remember to take 5% of 25000.

Problem #4:

You personal trainer tells you that your weight loss should be between 35 and 45 pounds to win a free training session. Write an absolute value inequality that model your weight loss.

The trick is to understand the meaning of absolute value in terms of distance .

|x - a|  = d

What does the absolute value equality above mean? It means to find 2 numbers that are located at the same distance d from a.

For example, |x - 1| = 3

This means to find two numbers that are 3 units away from 1. These 2 numbers are -2 and 4 of course since 4 - 1 = 3 and 1 - -2 = 3

If you are given -2 and 4, you will need to work backward to find 1 and 3.

The number that  -2 and 4  are the same distance from is the number between -2 and 4. To find the number that is between -2 and 4, taking the average of -2 and 4 will suffice. 

To find the distance, just do 4 - 1 = 3

We can do the same thing to find the absolute value inequality for 35 and 45

Since 45 - 40 = 5, the distance is 5.

The inequality is |x - 40| < 5

Solving absolute value inequalities

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Word Problems with Absolute Value Inequalities - Expii

Word problems with absolute value inequalities, explanations (3).

solving absolute value inequalities word problems worksheet

(Video) Absolute Value Inequality Word Problems

solving absolute value inequalities word problems worksheet

Here's a really good video by I ♥ π . She shows you how to systematically work through absolute inequality word problems.

To start off, she looks at translating word problems into mathematical expressions . A lot of absolute value inequality word problems have to do with the margin of error . In these, you want the difference between an unknown value and a goal value (that you know) to be less than a number. Watch the beginning of the video to find out how to set up that equation, then try the following problem.

It takes Jamil 40 minutes to get to work, plus or minus at most 8 minutes depending on traffic. If she leaves her house at 10:02, when's earliest she can expect to get in?

First, show find the range of times that it could take Jamil to get to work using an absolute value equation. Which of the following equations would work?

Related Lessons

To be able to solve Word Problems with Absolute Value Inequalities, you first need to be able to translate word problems into mathematical expressions . It would also be helpful to understand how to solve inequalities , in general. It may be helpful to to review solving word problems with inequalities .

Word problems with absolute value inequalities often talk about absolute error or tolerance .

If you see any of these phrases, that's a good indicator that you need to translate the word problem into an inequality using absolute values.

Let's look at an example:

Rochelle builds chairs for fun. Before she can sell the, each chair must measure 80 cm in height, with no more than 2.5 cm error. Write and solve an absolute value inequality for this situation.

Image source: by Hannah Bonville

We know that the error (the difference between the actual height of the chair and the target height of 80 cm) cannot be any more than 2.5 cm (which means the same thing as less than or equal to ).

We can write the inequality like this:

Recall that if our absolute value inequality uses < or ≤, we need to set up one inequality to solve.

What inequality represents this information?

−2.5≤x−80≤2.5

−5≤x−80≤2.5

solving absolute value inequalities word problems worksheet

When given a word problem relating to absolute value inequalities, first translate the words to math terms. In other words, represent the word problem as a mathematical equation or expression, and use variables to stand in for unknown quantities. Then, simply solve for the variables to solve the word problem!

Absolute value word problems often have phrases such as "give or take x units" or "margin of error" or "plus or minus x units" to let you know that the answers can be above OR below a given number.

Here is a graphic with an example!

solving absolute value inequalities word problems worksheet

Image source: By Caroline Kulczycky

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Absolute Value Inequalities Worksheets

Gravitate toward our printable absolute value inequalities worksheets if you want your grade 8 and high school students to steer clear through the two basic approaches in solving absolute value inequalities: graphically and algebraically. Solving an absolute inequality means finding the set of values that satisfies the inequality. The result can be an interval with overlapping values or the union of two intervals putting together the non-overlapping elements. Give a new lease of life to your practice with our free absolute value inequalities worksheets.

Solving Absolute Value Inequalities | Basic

Solving Absolute Value Inequalities

Clear the absolute-value bars by splitting the inequality into two pieces. Instruct grade 8 students to find the boundary points by solving each basic, one-step, and two-step absolute value inequality in this batch of printable worksheets.

  • Download the set

Absolute Value Inequalities | Basic

Solving and Graphing Absolute Value Inequalities

Students in 8th grade evaluate the inequality twice to account for both the positive and negative possibilities and graph the absolute value inequality as a segment between two points or as two rays going in the opposite directions.

Identifying Solution Graphs | Basic

Identifying Solution Graphs for Absolute Inequalities

Figure out the range of possible solutions in these absolute value inequalities worksheet pdfs. Check the solution graph that is inclusive or strict, and a segment between two points or a graph with two rays heading in the opposite directions based on your solution.

Identifying the Solution | Basic

Identifying the Solution

Get rid of the absolute value symbol, solve the inequalities as two linear inequalities, isolate the variable using the additive and multiplicative inverses and choose the option that best describes the solution in these high school absolute value inequalities worksheet pdfs.

Absolute Value Inequalities - Interval Notation | Basic

Choosing the Correct Interval Notation

High school students break each absolute inequality in two, solve both, and choose the solution presented as an interval notation with "and" or intersection symbol for "less than" inequality and "or" or union symbol for "greater than" inequality.

Related Worksheets

» Multi Step Inequalities

» Compound Inequalities

» Graphing Linear Inequalities

» Quadratic Inequalities

» Two Step Inequalities

» Absolute Value

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Mathematics LibreTexts

2.6: Solving Absolute Value Equations and Inequalities

  • Last updated
  • Save as PDF
  • Page ID 6238

Learning Objectives

  • Review the definition of absolute value.
  • Solve absolute value equations.
  • Solve absolute value inequalities.

Absolute Value Equations

Recall that the absolute value 63 of a real number \(a\), denoted \(|a|\), is defined as the distance between zero (the origin) and the graph of that real number on the number line. For example, \(|−3|=3\) and \(|3|=3\).

b4aaf730a9b49a73a73f1eee65bc4cc1.png

In addition, the absolute value of a real number can be defined algebraically as a piecewise function.

\(| a | = \left\{ \begin{array} { l } { a \text { if } a \geq 0 } \\ { - a \text { if } a < 0 } \end{array} \right.\)

Given this definition, \(|3| = 3\) and \(|−3| = − (−3) = 3\).Therefore, the equation \(|x| = 3\) has two solutions for \(x\), namely \(\{±3\}\). In general, given any algebraic expression \(X\) and any positive number \(p\):

\(\text{If}\: | X | = p \text { then } X = - p \text { or } X = p\)

In other words, the argument of the absolute value 64 \(X\) can be either positive or negative \(p\). Use this theorem to solve absolute value equations algebraically.

Example \(\PageIndex{1}\):

Solve: \(|x+2|=3\).

In this case, the argument of the absolute value is \(x+2\) and must be equal to \(3\) or \(−3\).

4a781f54e79090a87431680588379e8f.png

Therefore, to solve this absolute value equation, set \(x+2\) equal to \(±3\) and solve each linear equation as usual.

\(\begin{array} { c } { | x + 2 | = 3 } \\ { x + 2 = - 3 \quad \quad\text { or } \quad\quad x + 2 = 3 } \\ { x = - 5 \quad\quad\quad\quad\quad\quad\quad x = 1 } \end{array}\)

The solutions are \(−5\) and \(1\).

To visualize these solutions, graph the functions on either side of the equal sign on the same set of coordinate axes. In this case, \(f (x) = |x + 2|\) is an absolute value function shifted two units horizontally to the left, and \(g (x) = 3\) is a constant function whose graph is a horizontal line. Determine the \(x\)-values where \(f (x) = g (x)\).

8d0f6d91a1a35018cf3b4c0ce4b152da.png

From the graph we can see that both functions coincide where \(x = −5\) and \(x = 1\). The solutions correspond to the points of intersection.

Example \(\PageIndex{2}\):

Solve: \(| 2 x + 3 | = 4\).

Here the argument of the absolute value is \(2x+3\) and can be equal to \(-4\) or \(4\).

\(\begin{array} { r l } { | 2 x + 3 | } & { = \quad 4 } \\ { 2 x + 3 = - 4 } & { \text { or }\quad 2 x + 3 = 4 } \\ { 2 x = - 7 } & \quad\quad\:\: { 2 x = 1 } \\ { x = - \frac { 7 } { 2 } } & \quad\quad\:\: { x = \frac { 1 } { 2 } } \end{array}\)

Check to see if these solutions satisfy the original equation.

The solutions are \(-\frac{7}{2}\) and \(\frac{1}{2}\).

To apply the theorem, the absolute value must be isolated. The general steps for solving absolute value equations are outlined in the following example.

Example \(\PageIndex{3}\):

Solve: \(2 |5x − 1| − 3 = 9\).

Step 1 : Isolate the absolute value to obtain the form \(|X| = p\).

\(\begin{aligned} 2 | 5 x - 1 | - 3 & = 9 \:\:\:\color{Cerulean} { Add\: 3\: to\: both\: sides. } \\ 2 | 5 x - 1 | & = 12 \:\:\color{Cerulean} { Divide\: both\: sides\: by\: 2 } \\ | 5 x - 1 | & = 6 \end{aligned}\)

Step 2 : Set the argument of the absolute value equal to \(±p\). Here the argument is \(5x − 1\) and \(p = 6\).

\(5 x - 1 = - 6 \text { or } 5 x - 1 = 6\)

Step 3 : Solve each of the resulting linear equations.

\(\begin{array} { r l } { 5 x - 1 = - 6 \quad\:\:\text { or } \quad\quad5 x - 1 } & { \:\:\:\:\:\:= 6 } \\ { 5 x = - 5 }\quad\:\quad\quad\quad\quad\quad\quad\: & { 5 x = 7 } \\ { x = - 1 } \quad\quad\quad\quad\quad\quad\quad\:\:& { x = \frac { 7 } { 5 } } \end{array}\)

Step 4 : Verify the solutions in the original equation.

The solutions are \(-1\) and \(\frac{7}{5}\)

Exercise \(\PageIndex{1}\)

Solve: \(2 - 7 | x + 4 | = - 12\).

www.youtube.com/v/G0EjbqreYmU

Not all absolute value equations will have two solutions.

Example \(\PageIndex{4}\):

Solve: \(| 7 x - 6 | + 3 = 3\).

Begin by isolating the absolute value.

\(\begin{array} { l } { | 7 x - 6 | + 3 = 3 \:\:\:\color{Cerulean} { Subtract\: 3\: on\: both\: sides.} } \\ { \quad | 7 x - 6 | = 0 } \end{array}\)

Only zero has the absolute value of zero, \(|0| = 0\). In other words, \(|X| = 0\) has one solution, namely \(X = 0\). Therefore, set the argument \(7x − 6\) equal to zero and then solve for \(x\).

\(\begin{aligned} 7 x - 6 & = 0 \\ 7 x & = 6 \\ x & = \frac { 6 } { 7 } \end{aligned}\)

Geometrically, one solution corresponds to one point of intersection.

8cfd0197857b87c9691270321c077d08.png

The solution is \(\frac{6}{7}\).

Example \(\PageIndex{5}\):

Solve: \(|x+7|+5=4\).

\(\begin{aligned} | x + 7 | + 5 & = 4 \:\:\color{Cerulean} { Subtract \: 5\: on\: both\: sides.} \\ | x + 7 | & = - 1 \end{aligned}\)

In this case, we can see that the isolated absolute value is equal to a negative number. Recall that the absolute value will always be positive. Therefore, we conclude that there is no solution. Geometrically, there is no point of intersection.

f800ff00fb0c0579741140bd63f30a6c.png

There is no solution, \(Ø\).

If given an equation with two absolute values of the form \(| a | = | b |\), then \(b\) must be the same as \(a\) or opposite. For example, if \(a=5\), then \(b = \pm 5\) and we have:

\(| 5 | = | - 5 | \text { or } | 5 | = | + 5 |\)

In general, given algebraic expressions \(X\) and \(Y\):

\(\text{If} | X | = | Y | \text { then } X = - Y \text { or } X = Y\).

In other words, if two absolute value expressions are equal, then the arguments can be the same or opposite.

Example \(\PageIndex{6}\):

Solve: \(| 2 x - 5 | = | x - 4 |\).

Set \(2x-5\) equal to \(\pm ( x - 4 )\) and then solve each linear equation.

\(\begin{array} { c } { | 2 x - 5 | = | x - 4 | } \\ { 2 x - 5 = - ( x - 4 ) \:\: \text { or }\:\: 2 x - 5 = + ( x - 4 ) } \\ { 2 x - 5 = - x + 4 }\quad\quad\quad 2x-5=x-4 \\ { 3 x = 9 }\quad\quad\quad\quad\quad\quad \quad\quad x=1 \\ { x = 3 \quad\quad\quad\quad\quad\quad\quad\quad\quad\:\:\:\:} \end{array}\)

To check, we substitute these values into the original equation.

As an exercise, use a graphing utility to graph both \(f(x)= |2x-5|\) and \(g(x)=|x-4|\) on the same set of axes. Verify that the graphs intersect where \(x\) is equal to \(1\) and \(3\).

The solutions are \(1\) and \(3\).

Exercise \(\PageIndex{2}\)

Solve: \(| x + 10 | = | 3 x - 2 |\).

www.youtube.com/v/CskWmsQCBMU

Absolute Value Inequalities

We begin by examining the solutions to the following inequality:

\(| x | \leq 3\)

The absolute value of a number represents the distance from the origin. Therefore, this equation describes all numbers whose distance from zero is less than or equal to \(3\). We can graph this solution set by shading all such numbers.

54d7fb08641f26cbbcabbdf76cd2ada7.png

Certainly we can see that there are infinitely many solutions to \(|x|≤3\) bounded by \(−3\) and \(3\). Express this solution set using set notation or interval notation as follows:

\(\begin{array} { c } { \{ x | - 3 \leq x \leq 3 \} \color{Cerulean} { Set\: Notation } } \\ { [ - 3,3 ] \quad \color{Cerulean}{ Interval \:Notation } } \end{array}\)

In this text, we will choose to express solutions in interval notation. In general, given any algebraic expression \(X\) and any positive number \(p\):

\(\text{If} | X | \leq p \text { then } - p \leq X \leq p\).

This theorem holds true for strict inequalities as well. In other words, we can convert any absolute value inequality involving " less than " into a compound inequality which can be solved as usual.

Example \(\PageIndex{7}\):

Solve and graph the solution set: \(|x+2|<3\).

Bound the argument \(x+2\) by \(−3\) and \(3\) and solve.

\(\begin{array} { c } { | x + 2 | < 3 } \\ { - 3 < x + 2 < 3 } \\ { - 3 \color{Cerulean}{- 2}\color{Black}{ <} x + 2 \color{Cerulean}{- 2}\color{Black}{ <} 3 \color{Cerulean}{- 2} } \\ { - 5 < x < 1 } \end{array}\)

Here we use open dots to indicate strict inequalities on the graph as follows.

d198f40ad4498592e3fb836345d5d10d.png

Using interval notation, \((−5,1)\).

The solution to \(| x + 2 | < 3\) can be interpreted graphically if we let \(f ( x ) = | x + 2 |\) and \(g(x)=3\) and then determine where \(f ( x ) < g ( x )\) by graphing both \(f\) and \(g\) on the same set of axes.

2f9d712a7228e5ab4d7b725b4b5dd02a.png

The solution consists of all \(x\)-values where the graph of \(f\) is below the graph of \(g\). In this case, we can see that \(|x + 2| < 3\) where the \(x\)-values are between \(−5\) and \(1\). To apply the theorem, we must first isolate the absolute value.

Example \(\PageIndex{8}\):

Solve: \(4 |x + 3| − 7 ≤ 5\).

\(\begin{array} { c } { 4 | x + 3 | - 7 \leq 5 } \\ { 4 | x + 3 | \leq 12 } \\ { | x + 3 | \leq 3 } \end{array}\)

Next, apply the theorem and rewrite the absolute value inequality as a compound inequality.

\(\begin{array} { c } { | x + 3 | \leq 3 } \\ { - 3 \leq x + 3 \leq 3 } \end{array}\)

\(\begin{aligned} - 3 \leq x + 3 \leq & 3 \\ - 3 \color{Cerulean}{- 3} \color{Black}{ \leq} x + 3 \color{Cerulean}{- 3} & \color{Black}{ \leq} 3 \color{Cerulean}{- 3} \\ - 6 \leq x \leq 0 \end{aligned}\)

Shade the solutions on a number line and present the answer in interval notation. Here we use closed dots to indicate inclusive inequalities on the graph as follows:

8abae6a57d4658e1e28b81d29a8ba467.png

Using interval notation, \([−6,0]\)

Exercise \(\PageIndex{3}\)

Solve and graph the solution set: \(3 + | 4 x - 5 | < 8\).

Interval notation: \((0, \frac{5}{2})\)

53fbc057eea3477b5709c94560299f26.png

www.youtube.com/v/sX6ppL2Fbq0

Next, we examine the solutions to an inequality that involves " greater than ," as in the following example:

\(| x | \geq 3\)

This inequality describes all numbers whose distance from the origin is greater than or equal to \(3\). On a graph, we can shade all such numbers.

ee5443c82bae2d950182100b7ab9ff13.png

There are infinitely many solutions that can be expressed using set notation and interval notation as follows:

\(\begin{array} { l } { \{ x | x \leq - 3 \text { or } x \geq 3 \} \:\:\color{Cerulean} { Set\: Notation } } \\ { ( - \infty , - 3 ] \cup [ 3 , \infty ) \:\:\color{Cerulean} { Interval\: Notation } } \end{array}\)

In general, given any algebraic expression \(X\) and any positive number \(p\):

\(\text{If} | X | \geq p \text { then } X \leq - p \text { or } X \geq p\).

The theorem holds true for strict inequalities as well. In other words, we can convert any absolute value inequality involving “ greater than ” into a compound inequality that describes two intervals.

Example \(\PageIndex{9}\):

Solve and graph the solution set: \(|x+2|>3\).

The argument \(x+2\) must be less than \(−3\) or greater than \(3\).

\(\begin{array} { c } { | x + 2 | > 3 } \\ { x + 2 < - 3 \quad \text { or } \quad x + 2 > 3 } \\ { x < - 5 }\quad\quad\quad\quad\quad\: x>1 \end{array}\)

12b84b5b3331a000842bb1191680226e.png

Using interval notation, \((−∞,−5)∪(1,∞)\).

The solution to \(|x + 2| > 3\) can be interpreted graphically if we let \(f (x) = |x + 2|\) and \(g (x) = 3\) and then determine where \(f(x) > g (x)\) by graphing both \(f\) and \(g\) on the same set of axes.

solving absolute value inequalities word problems worksheet

The solution consists of all \(x\)-values where the graph of \(f\) is above the graph of \(g\). In this case, we can see that \(|x + 2| > 3\) where the \(x\)-values are less than \(−5\) or are greater than \(1\). To apply the theorem we must first isolate the absolute value.

Example \(\PageIndex{10}\):

Solve: \(3 + 2 |4x − 7| ≥ 13\).

\(\begin{array} { r } { 3 + 2 | 4 x - 7 | \geq 13 } \\ { 2 | 4 x - 7 | \geq 10 } \\ { | 4 x - 7 | \geq 5 } \end{array}\)

\(\begin{array} &\quad\quad\quad\quad\:\:\:|4x-7|\geq 5 \\ 4 x - 7 \leq - 5 \quad \text { or } \quad 4 x - 7 \geq 5 \end{array}\)

\(\begin{array} { l } { 4 x - 7 \leq - 5 \text { or } 4 x - 7 \geq 5 } \\ \quad\:\:\:\:{ 4 x \leq 2 } \quad\quad\quad\:\:\: 4x\geq 12\\ \quad\:\:\:\:{ x \leq \frac { 2 } { 4 } } \quad\quad\quad\quad x\geq 3 \\ \quad\quad{ x \leq \frac { 1 } { 2 } } \end{array}\)

Shade the solutions on a number line and present the answer using interval notation.

06fafde1bcd818b99ee494ab4c092fc0.png

Using interval notation, \((−∞,\frac { 1 } { 2 }]∪[3,∞)\)

Exercise \(\PageIndex{4}\)

Solve and graph: \(3 | 6 x + 5 | - 2 > 13\).

Using interval notation, \(\left( - \infty , - \frac { 5 } { 3 } \right) \cup ( 0 , \infty )\)

8775b7be1a775881d4ce3e89ac626c6c.png

www.youtube.com/v/P6HjRz6W4F4

Up to this point, the solution sets of linear absolute value inequalities have consisted of a single bounded interval or two unbounded intervals. This is not always the case.

Example \(\PageIndex{11}\):

Solve and graph: \(|2x−1|+5>2\).

\(\begin{array} { c } { | 2 x - 1 | + 5 > 2 } \\ { | 2 x - 1 | > - 3 } \end{array}\)

Notice that we have an absolute value greater than a negative number. For any real number x the absolute value of the argument will always be positive. Hence, any real number will solve this inequality.

1ae423963434b72a27df0ffb19ce6b24.png

Geometrically, we can see that \(f(x)=|2x−1|+5\) is always greater than \(g(x)=2\).

4df21f5c6ca04b0cb00c064214bbf386.png

All real numbers, \(ℝ\).

Example \(\PageIndex{12}\):

Solve and graph: \(|x+1|+4≤3\).

\(\begin{array} { l } { | x + 1 | + 4 \leq 3 } \\ { | x + 1 | \leq - 1 } \end{array}\)

In this case, we can see that the isolated absolute value is to be less than or equal to a negative number. Again, the absolute value will always be positive; hence, we can conclude that there is no solution.

Geometrically, we can see that \(f(x)=|x+1|+4\) is never less than \(g(x)=3\).

611732db35c3bdba71871ef237256092.png

Answer : \(Ø\)

In summary, there are three cases for absolute value equations and inequalities. The relations \(=, <, \leq, > \) and \(≥\) determine which theorem to apply.

Case 1: An absolute value equation:

Case 2: an absolute value inequality involving " less than .", case 3: an absolute value inequality involving " greater than .", key takeaways.

  • To solve an absolute value equation, such as \(|X| = p\), replace it with the two equations \(X = −p\) and \(X = p\) and then solve each as usual. Absolute value equations can have up to two solutions.
  • To solve an absolute value inequality involving “less than,” such as \(|X| ≤ p\), replace it with the compound inequality \(−p ≤ X ≤ p\) and then solve as usual.
  • To solve an absolute value inequality involving “greater than,” such as \(|X| ≥ p\), replace it with the compound inequality \(X ≤ −p\) or \(X ≥ p\) and then solve as usual.
  • Remember to isolate the absolute value before applying these theorems.

Exercise \(\PageIndex{5}\)

  • \(|x| = 9\)
  • \(|x| = 1\)
  • \(|x − 7| = 3\)
  • \(|x − 2| = 5\)
  • \(|x + 12| = 0\)
  • \(|x + 8| = 0\)
  • \(|x + 6| = −1\)
  • \(|x − 2| = −5\)
  • \(|2y − 1| = 13\)
  • \(|3y − 5| = 16\)
  • \(|−5t + 1| = 6\)
  • \(|−6t + 2| = 8\)
  • \(\left| \frac { 1 } { 2 } x - \frac { 2 } { 3 } \right| = \frac { 1 } { 6 }\)
  • \(\left| \frac { 2 } { 3 } x + \frac { 1 } { 4 } \right| = \frac { 5 } { 12 }\)
  • \(|0.2x + 1.6| = 3.6\)
  • \(|0.3x − 1.2| = 2.7\)
  • \(| 5 (y − 4) + 5| = 15\)
  • \(| 2 (y − 1) − 3y| = 4\)
  • \(|5x − 7| + 3 = 10\)
  • \(|3x − 8| − 2 = 6\)
  • \(9 + |7x + 1| = 9\)
  • \(4 − |2x − 3| = 4\)
  • \(3 |x − 8| + 4 = 25\)
  • \(2 |x + 6| − 3 = 17\)
  • \(9 + 5 |x − 1| = 4\)
  • \(11 + 6 |x − 4| = 5\)
  • \(8 − 2 |x + 1| = 4\)
  • \(12 − 5 |x − 2| = 2\)
  • \(\frac{1}{2} |x − 5| − \frac{2}{3} = −\frac{1}{6}\)
  • \(\frac { 1 } { 3 } \left| x + \frac { 1 } { 2 } \right| + 1 = \frac { 3 } { 2 }\)
  • \(−2 |7x + 1| − 4 = 2\)
  • \(−3 |5x − 3| + 2 = 5\)
  • \(1.2 |t − 2.8| − 4.8 = 1.2\)
  • \(3.6 | t + 1.8| − 2.6 = 8.2\)
  • \(\frac{1}{2} |2 (3x − 1) − 3| + 1 = 4\)
  • \(\frac{2}{3} |4 (3x + 1) − 1| − 5 = 3\)
  • \(|5x − 7| = |4x − 2|\)
  • \(|8x − 3| = |7x − 12|\)
  • \(|5y + 8| = |2y + 3|\)
  • \(|7y + 2| = |5y − 2|\)
  • \(|5 (x − 2)| = |3x|\)
  • \(|3 (x + 1)| = |7x|\)
  • \(\left| \frac { 2 } { 3 } x + \frac { 1 } { 2 } \right| = \left| \frac { 3 } { 2 } x - \frac { 1 } { 3 } \right|\)
  • \(\left| \frac { 3 } { 5 } x - \frac { 5 } { 2 } \right| = \left| \frac { 1 } { 2 } x + \frac { 2 } { 5 } \right|\)
  • \(|1.5t − 3.5| = |2.5t + 0.5|\)
  • \(|3.2t − 1.4| = |1.8t + 2.8|\)
  • \(|5 − 3 (2x + 1)| = |5x + 2|\)
  • \(|3 − 2 (3x − 2)| = |4x − 1|\)

1. \(−9, 9\)

3. \(4, 10\)

5. \(−12\)

7. \(Ø\)

9. \(−6, 7\)

11. \(−1, \frac{7}{5}\)

13. \(1, \frac{5}{3}\)

15. \(−26, 10\)

17. \(0, 6\)

19. \(0, \frac{14}{5}\)

21. \(−\frac{1}{7}\)

23. \(1, 15\)

25. \(Ø\)

27. \(−3, 1\)

29. \(4, 6\)

31. \(Ø\)

33. \(−2.2, 7.8\)

35. \(−\frac{1}{6}, \frac{11}{6}\)

37. \(1, 5\)

39. \(−\frac{5}{3}, −\frac{11}{7}\)

41. \(\frac{5}{4} , 5\)

43. \(−\frac{1}{13} , 1\)

45. \(−4, 0.75\)

47. \(0, 4\)

Exercise \(\PageIndex{6}\)

Solve and graph the solution set. In addition, give the solution set in interval notation.

  • Solve for \(x: p |ax + b| − q = 0\)
  • Solve for \(x: |ax + b| = |p + q|\)

1. \(x = \frac { - b q \pm q } { a p }\)

Exercise \(\PageIndex{7}\)

  • \(|x| < 5\)
  • \(|x| ≤ 2\)
  • \(|x + 3| ≤ 1\)
  • \(|x − 7| < 8\)
  • \(|x − 5| < 0\)
  • \(|x + 8| < −7\)
  • \(|2x − 3| ≤ 5\)
  • \(|3x − 9| < 27\)
  • \(|5x − 3| ≤ 0\)
  • \(|10x + 5| < 25\)
  • \(\left| \frac { 1 } { 3 } x - \frac { 2 } { 3 } \right| \leq 1\)
  • \(\left| \frac { 1 } { 12 } x - \frac { 1 } { 2 } \right| \leq \frac { 3 } { 2 }\)
  • \(|x| ≥ 5\)
  • \(|x| > 1\)
  • \(|x + 2| > 8\)
  • \(|x − 7| ≥ 11\)
  • \(|x + 5| ≥ 0\)
  • \(|x − 12| > −4\)
  • \(|2x − 5| ≥ 9\)
  • \(|2x + 3| ≥ 15\)
  • \(|4x − 3| > 9\)
  • \(|3x − 7| ≥ 2\)
  • \(\left| \frac { 1 } { 7 } x - \frac { 3 } { 14 } \right| > \frac { 1 } { 2 }\)
  • \(\left| \frac { 1 } { 2 } x + \frac { 5 } { 4 } \right| > \frac { 3 } { 4 }\)

1. \(( - 5,5 )\);

solving absolute value inequalities word problems worksheet

3. \([ - 4 , - 2 ]\);

solving absolute value inequalities word problems worksheet

5. \(\emptyset\);

18dc3829d035ad7a33a362ec1a7e5f09.png

7. \([ - 1,4 ]\);

4cb524885d5df3a2715ff992dcd5a6b3.png

9. \(\left\{ \frac { 3 } { 5 } \right\}\);

5fe987d3f662f3552c53cb4c800e585c.png

11. \([ - 1,5 ]\);

e698c903acbeaae5e83c1bc9c6e8845a.png

13. \(( - \infty , - 5 ] \cup [ 5 , \infty )\);

a7d47bfc737e19a60beecd34fdcbade7.png

15. \(( - \infty , - 10 ) \cup ( 6 , \infty )\);

9d1fb16fd22556532e81ef67b14c73a1.png

17. \(\mathbb { R }\);

6d8a25b4d71237af99c7a5f08d9d21e2.png

19. \(( - \infty , - 2 ] \cup [ 7 , \infty )\);

d1e745f298730b5b44f34dd56e033df2.png

21. \(\left( - \infty , - \frac { 3 } { 2 } \right) \cup ( 3 , \infty )\);

0010b88a922a03db183e099d487bc54b.png

23. \(( - \infty , - 2 ) \cup ( 5 , \infty )\);

33b71142f5fcc934441a9b670c22e110.png

Exercise \(\PageIndex{8}\)

Solve and graph the solution set.

  • \(|3 (2x − 1)| > 15\)
  • \(|3 (x − 3)| ≤ 21\)
  • \(−5 |x − 4| > −15\)
  • \(−3 |x + 8| ≤ −18\)
  • \(6 − 3 |x − 4| < 3\)
  • \(5 − 2 |x + 4| ≤ −7\)
  • \(6 − |2x + 5| < −5\)
  • \(25 − |3x − 7| ≥ 18\)
  • \(|2x + 25| − 4 ≥ 9\)
  • \(|3 (x − 3)| − 8 < −2\)
  • \(2 |9x + 5| + 8 > 6\)
  • \(3 |4x − 9| + 4 < −1\)
  • \(5 |4 − 3x| − 10 ≤ 0\)
  • \(6 |1 − 4x| − 24 ≥ 0\)
  • \(3 − 2 |x + 7| > −7\)
  • \(9 − 7 |x − 4| < −12\)
  • \(|5 (x − 4) + 5| > 15\)
  • \(|3 (x − 9) + 6| ≤ 3\)
  • \(\left| \frac { 1 } { 3 } ( x + 2 ) - \frac { 7 } { 6 } \right| - \frac { 2 } { 3 } \leq - \frac { 1 } { 6 }\)
  • \(\left| \frac { 1 } { 10 } ( x + 3 ) - \frac { 1 } { 2 } \right| + \frac { 3 } { 20 } > \frac { 1 } { 4 }\)
  • \(12 + 4 |2x − 1| ≤ 12\)
  • \(3 − 6 |3x − 2| ≥ 3\)
  • \(\frac{1}{2} |2x − 1| + 3 < 4\)
  • 2 |\frac{1}{2} x + \frac{2}{3} | − 3 ≤ −1\)
  • \(7 − |−4 + 2 (3 − 4x)| > 5\)
  • \(9 − |6 + 3 (2x − 1)| ≥ 8\)
  • \(\frac { 3 } { 2 } - \left| 2 - \frac { 1 } { 3 } x \right| < \frac { 1 } { 2 }\)
  • \(\frac { 5 } { 4 } - \left| \frac { 1 } { 2 } - \frac { 1 } { 4 } x \right| < \frac { 3 } { 8 }\)

1. \(( - \infty , - 2 ) \cup ( 3 , \infty )\);

ebcbc7e9a8c69eaf7dc93e0a895be02a.png

3. \(( 1,7 )\);

6a84ac9f29153422122cecb7ae173a27.png

5. \(( - \infty , 3 ) \cup ( 5 , \infty )\);

291445ead16c4e9f28633d76d92ebd51.png

7. \(( - \infty , - 8 ) \cup ( 3 , \infty )\);

22b7aed415f504f8381f18d4e7776f45.png

9. \(( - \infty , - 19 ] \cup [ - 6 , \infty )\);

13fa6f328decfd285cbe4425b06e4c93.png

11. \(\mathbb { R }\);

13. \(\left[ \frac { 2 } { 3 } , 2 \right]\);

ceec08d41c2361f4613cd5e5d213844e.png

15. \(( - 12 , - 2 )\);

9964b1ed11f681d97f2fa612af08d243.png

17. \(( - \infty , 0 ) \cup ( 6 , \infty )\);

755b3ba705f6729a468c1d53bd54200a.png

19. \([ 0,3 ]\);

1a67f6f2059288d7eb289b5b91888451.png

21. \(\frac { 1 } { 2 }\);

97abc5e7029c6c2abb111a8d6172fc3f.png

23. \(\left( - \frac { 1 } { 2 } , \frac { 3 } { 2 } \right)\);

9940bee4840619748881b6ee166fb60c.png

25. \(\left( 0 , \frac { 1 } { 2 } \right)\);

629e476a686ff5681e41138257e2f57e.png

27. \(( - \infty , 3 ) \cup ( 9 , \infty )\);

6f9be38567ebfe5e3669e0e423e0f33b.png

Exercise \(\PageIndex{9}\)

Assume all variables in the denominator are nonzero.

  • Solve for \(x\) where \(a, p > 0: p |ax + b| − q ≤ 0\)
  • Solve for \(x\) where \(a, p > 0: p |ax + b| − q ≥ 0\)

1. \(\frac { - q - b p } { a p } \leq x \leq \frac { q - b p } { a p }\)

Exercise \(\PageIndex{10}\)

Given the graph of \(f\) and \(g\), determine the \(x\)-values where:

(a) \(f ( x ) = g ( x )\)

(b) \(f ( x ) > g ( x )\)

(c) \(f ( x ) < g ( x )\)

bc0d2fc192eab76856e471f5ac12e232.png

1. (a) \(−6, 0\); (b) \((−∞, −6) ∪ (0, ∞)\); (c) \((−6, 0)\)

3. (a) \(Ø\); (b) \(ℝ\); (c) \(Ø\)

Exercise \(\PageIndex{11}\)

  • Make three note cards, one for each of the three cases described in this section. On one side write the theorem, and on the other write a complete solution to a representative example. Share your strategy for identifying and solving absolute value equations and inequalities on the discussion board.
  • Make your own examples of absolute value equations and inequalities that have no solution, at least one for each case described in this section. Illustrate your examples with a graph.

1. Answer may vary

63 The distance from the graph of a number \(a\) to zero on a number line, denoted \(|a|\).

64 The number or expression inside the absolute value.

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Everything You Need to Know about the Absolute Value Inequalities Worksheets

Created: March 16, 2023

Last updated: June 7, 2023

An absolute value inequality is a type of inequality that has an absolute value sign with a variable inside. It involves an absolute value algebraic expression with variables. In simple terms, an absolute value inequality can either have one of the following forms or can be converted to any of them:

While this is a core topic in math, most young learners may struggle to get a grasp of it. So, how do you ease the process for them? It’s simple: try out our absolute value equations and inequalities worksheets.

Absolute Value Inequalities Worksheet Answers

Absolute Value Inequalities Worksheet Answers

Absolute Value Inequality Worksheet

Absolute Value Inequality Worksheet

Absolute Value Inequalities Worksheet Algebra 2 Answers

Absolute Value Inequalities Worksheet Algebra 2 Answers

Answer Key Absolute Value Inequalities Worksheet Answers

Answer Key Absolute Value Inequalities Worksheet Answers

Absolute Value Inequalities Practice Worksheet

Absolute Value Inequalities Practice Worksheet

Solve Absolute Value Inequalities Worksheet

Solve Absolute Value Inequalities Worksheet

About the Solving Absolute Value Equations and Inequalities Worksheet Answer Key

The solving absolute value inequalities worksheet is an excellent resource for teaching young learners about absolute value inequalities. This worksheet contains a wide range of fun math exercises that range from easy to more complex questions. It can help encourage students to read and think logically, rather than merely approach this topic at a shallow level.

With this worksheet, kids will be able to go beyond merely recognizing patterns and instead employ their analytical skills.

But it doesn’t end here. The absolute value equations and inequalities worksheet also comes with an answer key and detailed solutions to each problem, which is quite handy for self-paced learning, allowing young learners to test their accuracy and monitor progress.

At the end of the day, you’ll have an independent math whiz on your hands.

Math for Kids

Benefits of the Absolute Value Inequalities Word Problems Worksheet

Wondering why you should download this worksheet? Well, it’s quite simple. The absolute value inequalities worksheet comes with a wide range of perks designed to boost learning and equip a student with a comprehensive understanding of the topic.

For starters, it helps kids improve their accuracy, speed, and logical reasoning skills. It does this by offering them an opportunity to work with a wide range of math problems and build their math foundation. At the end of the day, kids will be able to solve complex exercises in less time.

Absolute Value Inequalities Worksheets PDF

Absolute Value Inequalities Word Problems Worksheet

Absolute Value Inequalities Worksheet

Solving Absolute Value Inequalities Worksheet Answers

Solving Absolute Value Inequalities Worksheet

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.

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Absolute Value Inequalities Worksheets

To complete the free printable worksheets given in this post, pupils need to solve linear absolute value inequalities and plot the solutions graphically.

solving absolute value inequalities word problems worksheet

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Absolute Value Word Problems Worksheets

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About These 15 Worksheets

Absolute value word problems worksheets are  designed to teach and practice the concept of absolute value in the context of real-world situations. Absolute value is a mathematical term that represents the distance of a number from zero on the number line, regardless of direction, so it is always a non-negative value. These worksheets typically include word problems that require students to think about and apply absolute value in various scenarios.

Types of problems you might expect to see on such worksheets include:

Comparisons of Measurements – Problems where students must find the absolute difference between two measurements, such as heights, distances, temperatures, or weights.

Financial Situations – Scenarios involving profit and loss, price differences, or account balance changes.

Sports Statistics – Situations comparing scores, times, or points between players or teams.

Scientific Data – Problems involving pH levels, altitude changes, or temperature variations where students determine the absolute change. Here’s an example of an absolute value word problem:

Problem: During a week of fundraising, the school’s soccer team raised $350 on Monday. Due to some unforeseen expenses, they ended up using $150 of the raised funds on Tuesday. What is the absolute value of the difference in the soccer team’s funds from Monday to Tuesday?

Solution: To find the absolute value of the difference in the team’s funds, we subtract the smaller number from the larger number and ignore the sign of the result:

Absolute Difference = |$350 – (-$150)| = |$350 + $150| = |$500| = $500

So, the absolute value of the difference in the soccer team’s funds from Monday to Tuesday is $500.

What Are Absolute Value Word Problems?

Absolute value word problems involve using the concept of absolute value to solve real-life situations or scenarios. Absolute value represents the distance of a number from zero on a number line, regardless of its direction. In word problems, absolute value is often used to represent a magnitude or a distance, and it is denoted by vertical bars (| |).

Absolute value word problems can cover various topics and contexts, such as distances, temperatures, time intervals, financial transactions, and more. These problems typically require determining the absolute value of a quantity or comparing absolute values to find a solution.

For example, here’s a simple absolute value word problem:

Problem: A football team gains or loses yardage on each play. On one play, the team loses 12 yards. On the next play, they gain 8 yards. What is the total yardage change?

Solution: To find the total yardage change, we need to calculate the absolute value of the sum of the yardage gained and lost. In this case, the team lost 12 yards and gained 8 yards. So, the total yardage change is |(-12) + 8| = |-4| = 4 yards. Therefore, the total yardage change is 4 yards.

Absolute value word problems can become more complex, involving multiple quantities, comparisons, or inequalities. However, the underlying principle remains the same—finding the absolute value of a quantity to represent a magnitude or distance in the problem context.

How to Solve These Types of Problems

Solving absolute value word problems involves understanding the concept of absolute value and applying it to real-life scenarios. Here’s a step-by-step guide to help you solve absolute value word problems:

Begin by thoroughly understanding the problem statement. Identify the information provided and what you’re being asked to find. Pay attention to keywords and phrases that indicate the use of absolute value, such as “distance,” “deviation,” or “magnitude.”

Identify the absolute value expression – Look for the specific quantity or expression within the problem that is enclosed within absolute value symbols (| |). This expression represents the magnitude or distance from zero.

Set up the equation – Create an equation based on the information provided in the problem. Consider two scenarios – one where the absolute value expression is positive and another where it is negative. Set up the equation accordingly.

Solve for both cases – Solve the equation for each case you identified in the previous step. Remove the absolute value symbols and isolate the variable to find its possible values.

Check your solutions – Substitute the potential solutions back into the original problem and evaluate if they satisfy the given conditions. For absolute value problems involving distances, ensure that the solutions make sense within the context of the problem. State the final solution or solutions based on the context of the problem. Depending on the problem, you may need to provide a single answer, a range of values, or a set of possible solutions.

Check for extraneous solutions – Sometimes, when solving absolute value equations, extraneous solutions may arise. These are solutions that do not satisfy the original problem conditions. Double-check your solutions and ensure they are valid.

This Skill In The Real World

Here are a few examples of situations where absolute value is used:

Distance and Displacement – Absolute value is commonly used to represent distances or displacements. For instance, if you are calculating the distance between two cities or the displacement of an object from its initial position, you would use absolute value to ensure the result is positive.

Temperature – Absolute value can be used in temperature-related problems. For example, if you need to determine the difference in temperature between two points, you would take the absolute value to obtain a positive value regardless of whether one temperature is higher or lower than the other.

Finance and Economics – Absolute value is relevant in finance and economics when dealing with measures of deviation or error. For instance, calculating the absolute value of the difference between an actual value and an expected value can help determine the magnitude of the deviation or error.

Optimization and Constraints – In optimization problems, where you aim to maximize or minimize a certain quantity, absolute value can be used to express constraints. For example, if you have a constraint that the difference between two variables must be less than a certain value, you would use absolute value to formulate the constraint.

Physics and Science – Absolute value is frequently used in physics and other sciences. It can represent quantities like velocity, acceleration, force, or electric charge, where direction may not be relevant, and only the magnitude matters.

What Types of Jobs and Careers Use This Skill?

Several professions and fields require regularly solving absolute value word problems as part of their job responsibilities. Here are a few careers where the use of absolute value is common:

  • Mathematics and Statistics – Mathematicians and statisticians often encounter absolute value word problems in their work. They use absolute value to analyze data, measure distances, calculate deviations, and solve various mathematical equations and inequalities.
  • Physics and Engineering – Professionals in physics and engineering regularly work with absolute value concepts. They apply absolute value when dealing with measurements, vectors, forces, distances, and deviations in various physical phenomena.
  • Economics and Finance – Economists and financial analysts frequently use absolute value in their work. They apply it to analyze deviations, calculate errors, measure differences, and determine the magnitude of changes in economic indicators, financial data, or investment returns.
  • Operations Research and Optimization – Professionals in operations research and optimization use absolute value to formulate constraints and objective functions in mathematical models. They often encounter absolute value word problems when optimizing processes, resource allocation, scheduling, or decision-making.
  • Data Analysis and Machine Learning – Data analysts and machine learning specialists encounter absolute value problems when working with datasets. They use absolute value to calculate error metrics, evaluate model performance, handle outliers, or measure distances between data points.
  • Geography and Navigation – Geographers, cartographers, and navigation experts may use absolute value when calculating distances between locations or analyzing spatial data. Absolute value helps measure the magnitude of differences or deviations in geographic coordinates.
  • Transportation and Logistics – Professionals in transportation and logistics rely on absolute value to solve problems related to routes, distances, and delivery times. They may use it to minimize deviations, calculate optimal paths, or evaluate the efficiency of transportation networks.

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

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Algebra and Pre-Algebra

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Word problems

Sometimes systems of equations can be used to model word problems.  Let’s jump straight to an example.

Example:   The school that Matt goes to is selling tickets to a choral performance.  On the first day of ticket sales the school sold 12 adult tickets and 3 student tickets for a total of $129.  The school took in $104 on the second day by selling 2 adult tickets and 6 student tickets.  Find the price of an adult ticket and the price of a student ticket.

Solution:   Let a be the price of an adult ticket, and let s represent the price of a student ticket.  On the first day of the performance the 12 adult tickets were sold at the price of a and 3 student tickets were sold at the price of s .  The sum of their sales was $129.  We can model this by

\(12a + 3s = 129\)

Using a similar reasoning, we can model the second day of sales by

\(2a + 6s = 104\)

Combining these two equations gives us a system that we can solve!  We use elimination:

\(12a + 3s = 129\) \(2a + 6s = 104\)

\( - 24a - 6s =  - 258\) \(2a + 6s = 104\)

\( - 22a =  - 154\)

That is, an adult ticket cost $7.  Then by substituting \(a = 7\) into the second equation, we have

\(2a + 6s = 104\) \(2\left( 7 \right) + 6s = 104\) \(14 + 6s = 104\) \(6s = 90\) \(s = 15\)

That is, a student ticket costs $15.

Another Example:   The senior class at High School A and High School B planned separate trips to the water park.  The senior class at High School A rented and filled 8 vans and 4 buses with 256 students.  High School B rented and filled 4 vans and 6 buses with 312 students.  Each van and each bus carried the same number of students.  How many students can a van carry?  How many students can a bus carry?

Solution:   Let v be the number of students a van can carry.  Let b be the number of students a bus can carry.  High School A’s situation can be modeled by

\(8v + 4b = 256\)

Similarly, High School B’s situation can be modeled by

\(4v + 6b = 312\)

We solve the system using elimination

\(8v + 4b = 256\) \(4v + 6b = 312\)

\(8v + 4b = 256\) \( - 8v - 12b =  - 624\)

\( - 8b =  - 368\)

That is, a bus can hold 46 students.  Substituting 46 into the first equation gives

\(8v + 4b = 256\) \(8v + 4\left( {46} \right) = 256\) \(8v + 184 = 256\) \(8v = 72\) \(v = 9\)

That is, each van can hold 9 students.

Below you can   download   some   free   math worksheets and practice.

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This free worksheet contains 10 assignments each with 24 questions with answers. Example of one question:

Systems-of-Equations-and-Inequalities-Word-problems-easy

Watch below how to solve this example:

Systems-of-Equations-and-Inequalities-Word-problems-medium

Systems-of-Equations-and-Inequalities-Word-problems-hard

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Absolute Value Inequality Worksheet 1 –  Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities.  These are one-step inequalities with mostly positive integers. Absolute Value Equations Worksheet 1 RTF Absolute Value Equations Worksheet 1 PDF View Answers

Absolute Value Inequality Worksheet 3 –  Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities.  These are two-step inequalities where you’ll need to use all of your inverse operations knowledge. Absolute Value Equations Worksheet 3 RTF Absolute Value Equations Worksheet 3 PDF View Answers

Absolute Value Inequality Worksheet 4 –  Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities.  These are two-step inequalities that can get quite complicated.  A nice challenge for your higher-level learners. Absolute Value Equations Worksheet 4 RTF Absolute Value Equations Worksheet 4 PDF View Answers

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  1. Absolute Value Word Problems Worksheet

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  2. Solving Absolute Value Equations & Inequalities Worksheet by Lexie

    solving absolute value inequalities word problems worksheet

  3. Absolute Value Inequalities Worksheets with Answer Key

    solving absolute value inequalities word problems worksheet

  4. Graphing Absolute Value Inequalities Worksheet

    solving absolute value inequalities word problems worksheet

  5. 30 Absolute Value Inequalities Worksheet

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  6. Absolute Value Inequalities Word Problems Worksheet With Ans

    solving absolute value inequalities word problems worksheet

VIDEO

  1. Absolute value word problems

  2. Solving Absolute Value Inequalities

  3. Absolute value inequalities

  4. Absolute Value Inequalities

  5. Solving Inequalities Interval Notation, Number Line, Absolute Value, Fractions & Variables

  6. Absolute Value Inequality Word Problems

COMMENTS

  1. PDF Absolute Value Word Problems Homework

    Absolute Value Word Problems 1) A machine is used to fill each of several bags with 16 ounces of sugar. After the bags are filled, another machine weighs them. If the bag weighs .3 ounces more or less than the desired weight, the bag is rejected.

  2. Absolute Value Word Problems

    Problem #1: Your have money in your wallet, but you don't know the exact amount. When a friend asks you, you say that you have 50 dollars give or take 15. Use an absolute value equation to find least and biggest amount of money in your pocket? Solution: Let x be the possible amount of money in your pocket. |x - 50| = 15 Equation #1 x - 50 = 15

  3. Algebra

    Here is a set of practice problems to accompany the Absolute Value Inequalities section of the Solving Equations and Inequalities chapter of the notes for Paul Dawkins Algebra course at Lamar University.

  4. Word Problems with Absolute Value Inequalities

    To be able to solve Word Problems with Absolute Value Inequalities, you first need to be able to translate word problems into mathematical expressions. It would also be helpful to understand how to solve inequalities, in general. It may be helpful to to review solving word problems with inequalities.

  5. PDF Solve each inequality and graph its solution.

    Absolute Value Inequalities Date_____ Period____ Solve each inequality and graph its solution. 1) n ... Create your own worksheets like this one with Infinite Algebra 2. Free trial available at KutaSoftware.com. Title: Absolute Value Inequalities.ks-ia2 Author: Mike

  6. Absolute value inequalities word problem (video)

    Solving absolute value inequalities 1 Solving absolute value inequalities 2 Solving absolute value inequalities: fractions Solving absolute value inequalities: no solution Absolute value inequalities word problem Math > Algebra (all content) > Absolute value equations, functions, & inequalities > Solving absolute value inequalities

  7. 5.5: Solve Absolute Value Inequalities (optional challenge practice)

    Solve Absolute Value Inequalities with "Less Than" Let's look now at what happens when we have an absolute value inequality. Everything we've learned about solving inequalities still holds, but we must consider how the absolute value impacts our work. Again we will look at our definition of absolute value.

  8. Absolute Value Inequalities Worksheets

    Solving Absolute Value Inequalities Clear the absolute-value bars by splitting the inequality into two pieces. Instruct grade 8 students to find the boundary points by solving each basic, one-step, and two-step absolute value inequality in this batch of printable worksheets. Basic One-Step Two-Step Download the set

  9. PDF Absolute Value Inequalities.ks-ia1

    Solve each inequality and graph its solution. 1) ... Absolute Value Inequalities Date_____ Period____ Solve each inequality and graph its solution. ... Create your own worksheets like this one with Infinite Algebra 1. Free trial available at KutaSoftware.com. Title: Absolute Value Inequalities.ks-ia1 Author:

  10. 2.6: Solving Absolute Value Equations and Inequalities

    Step 2: Set the argument of the absolute value equal to ± p. Here the argument is 5x − 1 and p = 6. 5x − 1 = − 6 or 5x − 1 = 6. Step 3: Solve each of the resulting linear equations. 5x − 1 = − 6 or 5x − 1 = 6 5x = − 5 5x = 7 x = − 1 x = 7 5. Step 4: Verify the solutions in the original equation. Check x = − 1.

  11. Algebra 1

    This algebra 1 worksheet will produce absolute value inequalities problems for graphing. You may select which type of inequality to use in the problems. Inequalities to Use in Problems Addition and Subtraction Inequalities Multiplication/Division Inequalities Mixture of Both Types of Inequalities Language for the Inequalities Worksheet

  12. A Handy Guide to Our Absolute Value Inequalities Worksheet [PDFs]

    The solving absolute value inequalities worksheet is an excellent resource for teaching young learners about absolute value inequalities. This worksheet contains a wide range of fun math exercises that range from easy to more complex questions.

  13. PDF Solving Absolute Value Equations

    ©E X2o051 2s nKZuut 5aw KS0ozf Tt0w 4a r8eq 4L qL WC5.M 9 lA kl fl B 9rZi Ugrh 5tXs8 XrKeZsieJrKvNezdQ.5 J hM Pazd leL iw 4iatTh 8 4ITnHfOiJn piwt2ev JA TlbgReub9rba K x2S.R Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name_____ Solving Absolute Value Equations Date_____ Period____

  14. Absolute Value Inequalities Worksheets with Answer Key

    Absolute Value Inequalities Worksheets. Tags: 4th Grade 5th Grade 6th Grade 7th Grade. To complete the free printable worksheets given in this post, pupils need to solve linear absolute value inequalities and plot the solutions graphically. Download PDF. Download PDF. Download PDF. Download PDF.

  15. Absolute value inequalities

    Whenever we have absolute value inequalities, we have to split the equation into a positive side and a negative side so that we account for both directions. You have to flip the inequality for the negative side. x ≥ 4 x ≥ 4 or x ≤ −4 x ≤ − 4. To graph this, we will use closed circles since the answer includes 4 and -4. Example 2:

  16. Absolute Value Word Problems Worksheets

    About These 15 Worksheets. Absolute value word problems worksheets are designed to teach and practice the concept of absolute value in the context of real-world situations. Absolute value is a mathematical term that represents the distance of a number from zero on the number line, regardless of direction, so it is always a non-negative value.

  17. PDF Solving Absolute Value Equations and Inequalities

    Steps for Solving Linear Absolute Value Equations: i.e. + = Isolate the absolute value. Identify what the isolated absolute value is set equal to... If the absolute value is set equal to zero, remove absolute value symbols & solve the equation to get one solution. If the absolute value is set equal to a negative number, there is no solution. c.

  18. Absolute Value Inequalities

    This is case 4. Example 3: Solve the absolute value inequality. This is a "less than" absolute value inequality which is an example of case 1. Get rid of the absolute value symbol by applying the rule. Then solve the linear inequality that arises. The goal is to isolate the variable " [latex]x [/latex]" in the middle.

  19. Absolute Value Inequalities Worksheets

    Absolute Value Inequality Worksheet 1 - Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities. These are one-step inequalities with mostly positive integers. Absolute Value Equations Worksheet 1 RTF Absolute Value Equations Worksheet 1 PDF View Answers

  20. Absolute Value Inequalities Worksheets

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

  21. Inequalities Word Problems Worksheets

    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.

  22. Word problems systems of equations and inequalities

    Below you can download some free math worksheets and practice. Systems-of-Equations-and-Inequalities-Word-problems-easy.pdf. Download. Downloads: 36256 x. This free worksheet contains 10 assignments each with 24 questions with answers. Example of one question: Watch below how to solve this example:

  23. Solving Inequalities with Absolute Value Worksheets

    Absolute Value Inequality Worksheet 1 - Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities. These are one-step inequalities with mostly positive integers. Absolute Value Equations Worksheet 1 RTF Absolute Value Equations Worksheet 1 PDF View Answers