# Absolute Value Function Worksheet - Page 2

Absolute Value Function Worksheet
• Page 2
11.
Find the values of $x$, if |4$x$ + 1| = $\frac{4}{3}$.
 a. $\frac{-1}{12}$, $\frac{7}{12}$ b. 1, -7 c. 1, $\frac{1}{12}$ d. $\frac{1}{12}$, $\frac{-7}{12}$

#### Solution:

|4x + 1| = 4 / 3

4x + 1 = 4 / 3 or 4x + 1 = -4 / 3
[The expression 4x + 1 is equal to -4 / 3 or 4 / 3.]

4x + 1 - 1 = 4 / 3 - 1 or 4x + 1 - 1 = -4 / 3 - 1
[Subtracting 1 from the two sides of the equation.]

4x = 1 / 3 or 4x = -7 / 3
[Simplify.]

x = 1 / 12 or x = -7 / 12
[Divide throughout by 4.]

The two solutions of the expression are 1 / 12 and -7 / 12.

12.
Find the values of $x$, if |4$x$ - 16| - 19 = 12.
 a. $\frac{47}{4}$, $\frac{47}{4}$ b. $\frac{47}{4}$, $\frac{-15}{4}$ c. $\frac{47}{4}$, $\frac{1}{4}$ d. $\frac{-47}{4}$, $\frac{-15}{4}$

#### Solution:

|4x - 16| - 19 = 12

|4x - 16| - 19 + 19 = 12 + 19
[Add 19 to both sides of the equation.]

|4x - 16| = 31
[Simplify.]

|4x - 16| = 31, 4x - 16 equals to 31 or - 31.

4x - 16 = 31 or 4x - 16 = - 31.

4x - 16 + 16 = 31 + 16 or 4x - 16 + 16 = - 31 + 16
[Add 16 to both sides of the equation.]

4x = 47 or 4x = -15
[Simplify.]

x = 47 / 4 or x = -15 / 4
[Divide throughout by 4.]

The equation has two solutions: 47 / 4 and -15 / 4.

13.
Find the values of $x$, if |11 - 6$x$| - $\frac{1}{4}$ = 5.6.
 a. -0.86, 2.81 b. 0.86, 2.81 c. 0.86, -2.81 d. -0.86, -2.81

#### Solution:

|11 - 6x| - 1 / 4 = 5.6

|11 - 6x| - 1 / 4 + 1 / 4 = 5.6 + 1 / 4
[Add 1 / 4 to both sides of the equation.]

|11 - 6x| = 5.6 + 0.25 = 5.85
[Simplify.]

|11 - 6x| = 5.85, 11 - 6x equals 5.85 or - 5.85.

11 - 6x = 5.85 or 11 - 6x = - 5.85

11 - 6x - 11 = 5.85 - 11 or 11 - 6x - 11 = - 5.85 - 11
[Subtracting 11 from the two sides of the equation.]

x = 0.86 or x = 2.81
[Simplify.]

The equation has two solutions: 0.86 and 2.81.

14.
Which of the following is an absolute value function?
 a. $f$($x$) = |8$x$ - 10| b. $f$($x$) = 10$x$3 - 8$x$ - 5 c. $f$($x$) = d. $f$($x$) = 10${e}^{8x+5}$

#### Solution:

f(x) = 10x3 - 8x - 5 is a cubic function.

f(x) = 10x - 5x+8 is a rational function.

f(x) = 10e8x+5 is an exponential function.

f(x) = |8x - 10| an absolute value function.

15.
0
 a. $b$ = - 9 or $b$ = - 14 b. $b$ = 4 or $b$ = 9 c. $b$ = - 4 or $b$ = - 14 d. $b$ = 4 or $b$ = - 14

#### Solution:

|b + 5| = 9

b + 5 = 9 or b + 5 = - 9

b = 4 or b = - 14

16.
0
 a. $a$ = 20 b. $a$ = 20 or $a$ = - 30 c. $a$ = 105 or $a$ = -151 d. $a$ = 29.2 or $a$ = -19.2

#### Solution:

|5a + 25| - 2 = 123

|5a + 25| = 125
[Add 2 to both sides of the equation.]

5a + 25 = 125 or 5a + 25 = - 125
[Express as a disjunction.]

5a = 100 or 5a = - 150
[Subtracting 25 from the two sides of the equation.]

a = 20 or a = - 30
[Divide throughout by 5 .]

17.
0
 a. $z$ = $\frac{6}{5}$ or $z$ = 6 b. $z$ = $\frac{33}{5}$or $z$ = - 6 c. $z$ = - $\frac{33}{5}$ or $z$ = - 44 d. $z$ = - $\frac{33}{5}$ or $z$ = 11

#### Solution:

|11 - 5z| = 44

11 - 5z = 44 or 11 - 5z = - 44

- 5z = 33 or - 5z = - 55
[Subtracting 11 from the two sides of the equation.]

z = - 33 / 5 or z = 11
[Divide throughout by 5.]

18.
Determine whether the statement is true or False: If $y$ < 0, then |6 - $y$| = 6 + $y$
 a. True b. False

#### Solution:

let y = - a, where a is any positive real number.
[y < 0.]

|6 - (- a)| = 6 - a
[Replace y = - a in the given statement.]

|6 + a| = 6 - a

6 + a = 6 - a, which is always false because a > 0.
[|x| = x, if x is positive.]

19.
Determine whether the given statement is true or false: If $x$ ≥ 0, then |$x$ - 4| = 4 - $x$
 a. True b. False

#### Solution:

If x ≥ 0, then |x - 4| = 4 - x

Take one example x = 8, which is greater than zero.

|8 - 4| = 4 - 8

4 = - 4, which is not true.

Hence, the given statement is not true.

20.
Determine whether the given statement is True or false: If $k$ ≤ 0, then |$k$ - 3| = |3 - $k$|
 a. False b. True

#### Solution:

Since k ≤ 0, let k = - b, where ' b ' is any positive real number.

|- b - 3| = |3 - (- b)|
[Substitute the values.]

|- (b + 3)| = |3 + b|

|b + 3| = |b + 3|, which is true.
[|- k| = |k|.]

So, the given statement is true.