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Access these online resources for additional instruction and practice with absolute value.

Key concepts

  • Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See [link] .
  • The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See [link] .
  • In an absolute value equation, an unknown variable is the input of an absolute value function.
  • If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See [link] .

Section exercises

Verbal

How do you solve an absolute value equation?

Isolate the absolute value term so that the equation is of the form | A | = B . Form one equation by setting the expression inside the absolute value symbol, A , equal to the expression on the other side of the equation, B . Form a second equation by setting A equal to the opposite of the expression on the other side of the equation, B . Solve each equation for the variable.

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How can you tell whether an absolute value function has two x -intercepts without graphing the function?

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When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

The graph of the absolute value function does not cross the x -axis, so the graph is either completely above or completely below the x -axis.

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How can you use the graph of an absolute value function to determine the x -values for which the function values are negative?

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Algebraic

Describe all numbers x that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.

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Describe all numbers x that are at a distance of 1 2 from the number −4. Express this set of numbers using absolute value notation.

| x + 4 | = 1 2

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Describe the situation in which the distance that point x is from 10 is at least 15 units. Express this set of numbers using absolute value notation.

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Find all function values f ( x ) such that the distance from f ( x ) to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.

| f ( x ) 8 | < 0.03

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For the following exercises, find the x - and y -intercepts of the graphs of each function.

f ( x ) = 4 | x 3 | + 4

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f ( x ) = 3 | x 2 | 1

( 0 , 7 ) ; no x -intercepts

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f ( x ) = 2 | x + 1 | + 6

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f ( x ) = 5 | x + 2 | + 15

( 0 , 5 ) , ( 1 , 0 ) , ( 5 , 0 )

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f ( x ) = 2 | x 1 | 6

( 0 , 4 ) , ( 4 , 0 ) , ( 2 , 0 )

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f ( x ) = | 2 x + 1 | 13

( 0 , 12 ) , ( 6 , 0 ) , ( 7 , 0 )

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f ( x ) = | x 9 | + 16

( 0 , 7 ) , ( 25 , 0 ) , ( 7 , 0 )

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Graphical

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

For the following exercises, graph the given functions by hand.

y = | x 3 | 2

Graph of an absolute function.
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f ( x ) = | x 1 | 2

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f ( x ) = | x + 3 | + 4

Graph of an absolute function.
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f ( x ) = 2 | x + 3 | + 1

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f ( x ) = 3 | x 2 | + 3

Graph of an absolute function.
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f ( x ) = | 2 x 4 | 3

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f ( x ) = | 3 x + 9 | + 2

Graph of an absolute function.
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f ( x ) = | x 1 | 3

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f ( x ) = | x + 4 | 3

Graph of an absolute function.
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f ( x ) = 1 2 | x + 4 | 3

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Technology

Use a graphing utility to graph f ( x ) = 10 | x 2 | on the viewing window [ 0 , 4 ] . Identify the corresponding range. Show the graph.

range: [ 0 , 20 ]

Graph of an absolute function.
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Use a graphing utility to graph f ( x ) = 100 | x | + 100 on the viewing window [ 5 , 5 ] . Identify the corresponding range. Show the graph.

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For the following exercises, graph each function using a graphing utility. Specify the viewing window.

f ( x ) = 0.1 | 0.1 ( 0.2 x ) | + 0.3

x - intercepts:

Graph of an absolute function.
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f ( x ) = 4 × 10 9 | x ( 5 × 10 9 ) | + 2 × 10 9

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Extensions

For the following exercises, solve the inequality.

If possible, find all values of a such that there are no x - intercepts for f ( x ) = 2 | x + 1 | + a .

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If possible, find all values of a such that there are no y -intercepts for f ( x ) = 2 | x + 1 | + a .

There is no solution for a that will keep the function from having a y -intercept. The absolute value function always crosses the y -intercept when x = 0.

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Real-world applications

Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and x represents the distance from city B to city A, express this using absolute value notation.

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The true proportion p of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

| p 0.08 | 0.015

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Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable x for the score.

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A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using x as the diameter of the bearing, write this statement using absolute value notation.

| x 5.0 | 0.01

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The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is x inches, express the tolerance using absolute value notation.

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Questions & Answers

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
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
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply

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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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