# 3.6 Absolute value functions  (Page 3/3)

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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] .

## Verbal

How do you solve an absolute value equation?

Isolate the absolute value term so that the equation is of the form $\text{\hspace{0.17em}}|A|=B.\text{\hspace{0.17em}}$ Form one equation by setting the expression inside the absolute value symbol, $\text{\hspace{0.17em}}A,\text{\hspace{0.17em}}$ equal to the expression on the other side of the equation, $\text{\hspace{0.17em}}B.\text{\hspace{0.17em}}$ Form a second equation by setting $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ equal to the opposite of the expression on the other side of the equation, $\text{\hspace{0.17em}}-B.\text{\hspace{0.17em}}$ Solve each equation for the variable.

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

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 $\text{\hspace{0.17em}}x$ -axis, so the graph is either completely above or completely below the $\text{\hspace{0.17em}}x$ -axis.

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

## Algebraic

Describe all numbers $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.

Describe all numbers $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that are at a distance of $\text{\hspace{0.17em}}\frac{1}{2}\text{\hspace{0.17em}}$ from the number −4. Express this set of numbers using absolute value notation.

$\text{\hspace{0.17em}}|x+4|=\frac{1}{2}\text{\hspace{0.17em}}$

Describe the situation in which the distance that point $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is from 10 is at least 15 units. Express this set of numbers using absolute value notation.

Find all function values $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ such that the distance from $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.

$|f\left(x\right)-8|<0.03$

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

$f\left(x\right)=4|x-3|+4$

$f\left(x\right)=-3|x-2|-1$

$\left(0,-7\right);\text{\hspace{0.17em}}$ no $\text{\hspace{0.17em}}x$ -intercepts

$f\left(x\right)=-2|x+1|+6$

$f\left(x\right)=-5|x+2|+15$

$\left(0,\text{\hspace{0.17em}}5\right),\left(1,0\right),\left(-5,0\right)$

$f\left(x\right)=2|x-1|-6$

$\left(0,-4\right),\left(4,0\right),\left(-2,0\right)$

$f\left(x\right)=|-2x+1|-13$

$\left(0,-12\right),\left(-6,0\right),\left(7,0\right)$

$f\left(x\right)=-|x-9|+16$

$\left(0,7\right),\left(25,0\right),\left(-7,0\right)$

## Graphical

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

$y=|x-1|$

$y=|x+1|$

$y=|x|+1$

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

$y=|x|-2$

$y=-|x|$

$y=-|x|-2$

$y=-|x-3|-2$

$f\left(x\right)=-|x-1|-2$

$f\left(x\right)=-|x+3|+4$

$f\left(x\right)=2|x+3|+1$

$f\left(x\right)=3|x-2|+3$

$f\left(x\right)=|2x-4|-3$

$f\left(x\right)=|3x+9|+2$

$f\left(x\right)=-|x-1|-3$

$f\left(x\right)=-|x+4|-3$

$f\left(x\right)=\frac{1}{2}|x+4|-3$

## Technology

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

range: $\text{\hspace{0.17em}}\left[0,20\right]$

Use a graphing utility to graph $\text{\hspace{0.17em}}f\left(x\right)=-100|x|+100\text{\hspace{0.17em}}$ on the viewing window $\text{\hspace{0.17em}}\left[-5,5\right].\text{\hspace{0.17em}}$ Identify the corresponding range. Show the graph.

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

$f\left(x\right)=-0.1|0.1\left(0.2-x\right)|+0.3$

$x\text{-}$ intercepts:

$f\left(x\right)=4×{10}^{9}|x-\left(5×{10}^{9}\right)|+2×{10}^{9}$

## Extensions

For the following exercises, solve the inequality.

If possible, find all values of $a$ such that there are no $x\text{-}$ intercepts for $f\left(x\right)=2|x+1|+a.$

If possible, find all values of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ such that there are no $\text{\hspace{0.17em}}y$ -intercepts for $\text{\hspace{0.17em}}f\left(x\right)=2|x+1|+a.$

There is no solution for $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ that will keep the function from having a $\text{\hspace{0.17em}}y$ -intercept. The absolute value function always crosses the $\text{\hspace{0.17em}}y$ -intercept when $\text{\hspace{0.17em}}x=0.$

## 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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represents the distance from city B to city A, express this using absolute value notation.

The true proportion $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ 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.

$|p-0.08|\le 0.015$

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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ for the score.

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ as the diameter of the bearing, write this statement using absolute value notation.

$|x-5.0|\le 0.01$

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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ inches, express the tolerance using absolute value notation.

12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8