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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

what are you up to?
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
hi
jai
hi Miranda
jai
thanks
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
lolllll if you say so
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
oh woww I understand
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
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
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
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
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

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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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