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−200 x 600 200 −200 + 600 x 600 + 600 200 + 600 400 x 800

This means our returns would be between $400 and $800.

To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.

Absolute value inequalities

For an algebraic expression X, and k > 0 , an absolute value inequality is an inequality of the form

| X | < k  is equivalent to  k < X < k | X | > k  is equivalent to  X < k  or  X > k

These statements also apply to | X | k and | X | k .

Determining a number within a prescribed distance

Describe all values x within a distance of 4 from the number 5.

We want the distance between x and 5 to be less than or equal to 4. We can draw a number line, such as in [link] , to represent the condition to be satisfied.

A number line with one tick mark in the center labeled: 5.  The tick marks on either side of the center one are not marked.  Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.

The distance from x to 5 can be represented using an absolute value symbol, | x 5 | . Write the values of x that satisfy the condition as an absolute value inequality.

| x 5 | 4

We need to write two inequalities as there are always two solutions to an absolute value equation.

x 5 4 and x 5 4 x 9 x 1

If the solution set is x 9 and x 1 , then the solution set is an interval including all real numbers between and including 1 and 9.

So | x 5 | 4 is equivalent to [ 1 , 9 ] in interval notation.

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Describe all x- values within a distance of 3 from the number 2.

| x −2 | 3

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Solving an absolute value inequality

Solve | x 1 | 3 .

| x 1 | 3 −3 x 1 3 −2 x 4 [ −2 , 4 ]
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Using a graphical approach to solve absolute value inequalities

Given the equation y = 1 2 | 4 x 5 | + 3 , determine the x -values for which the y -values are negative.

We are trying to determine where y < 0 , which is when 1 2 | 4 x 5 | + 3 < 0. We begin by isolating the absolute value.

1 2 | 4 x 5 | < 3 Multiply both sides by –2, and reverse the inequality . | 4 x 5 | > 6

Next, we solve for the equality | 4 x 5 | = 6.

4 x 5 = 6 4 x 5 = 6 4 x = 11 or 4 x = 1 x = 11 4 x = 1 4

Now, we can examine the graph to observe where the y- values are negative. We observe where the branches are below the x- axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at x = 1 4 and x = 11 4 , and that the graph opens downward. See [link] .

A coordinate plan with the x-axis ranging from -5 to 5 and the y-axis ranging from -4 to 4.  The function y = -1/2|4x – 5| + 3 is graphed.  An open circle appears at the point -0.25 and an arrow
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Solve 2 | k 4 | 6.

k 1 or k 7 ; in interval notation, this would be ( , 1 ] [ 7 , ) .

A coordinate plane with the x-axis ranging from -1 to 9 and the y-axis ranging from -3 to 8.  The function y = -2|k  4| + 6 is graphed and everything above the function is shaded in.
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Access these online resources for additional instruction and practice with linear inequalities and absolute value inequalities.

Key concepts

  • Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. See [link] and [link] .
  • Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. See [link] , [link] , [link] , and [link] .
  • Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities. See [link] and [link] .
  • Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. See [link] and [link] .
  • Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. See [link] .

Questions & Answers

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Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
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Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
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Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
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Miranda
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Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
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Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
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Chinni
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Chinni
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Hassan
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SORIE
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Abdel
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SORIE
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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