<< 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

(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
modul questions trigonometry
Thamarai Reply
(1+cosA)(1-cosA)=sin^2A
BINCY
differentiate f(t)=1/4t to the power 4 +8
Jessica Reply
I need trigonometry,polynomial
duru Reply
ok
Augustine
Why is 7 on top
Bertha Reply
simplify cot x / csc x
Catherine Reply
👉🌹Solve🌻 Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
what is the period of cos?
SIYAMTHEMBA Reply
your question might not seem clear as you asked. ask well to get perfect answers put your question on a table I'm willing to help you Mr Siyamthemba
Patrick
simplify: cot x/csc x
Catherine
sorry i didnt realize you were actually asking someone else to put their question on here. i thought this was where i was supposed to.
Catherine
some to dereve formula for bulky density
kurash
Solve Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
questions
Thamarai
ok
AjA
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
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