<< Chapter < Page Chapter >> Page >
−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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

| x −2 | 3

Got questions? Get instant answers now!

Solving an absolute value inequality

Solve | x 1 | 3 .

| x 1 | 3 −3 x 1 3 −2 x 4 [ −2 , 4 ]
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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

Find that number sum and product of all the divisors of 360
jancy Reply
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
madras university algebra questions papers first year B. SC. maths
Kanniyappan Reply
Hey
Rightspect
hi
chesky
Give me algebra questions
Rightspect
how to send you
Vandna
What does this mean
Michael Reply
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
rajan Reply
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
Ching Reply
give me some important question in tregnamentry
Anshuman
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask