# Absolute Value Function Worksheet - Page 4

Absolute Value Function Worksheet
• Page 4
31.
Solve for $x$ and check: |$x$ - 2| = 3$x$ + 4$a$, $a$ is a positive real number.
 a. No solution b. $x$ = (- 1 - 2$a$) only c. $x$ = (- 1 - 2$a$) or $x$ = ( $\frac{1}{2}$ - $a$) d. $x$ = ( $\frac{1}{2}$ - $a$) only

#### Solution:

|x - 2| = 3x + 4a

x - 2 = 3x + 4a or x - 2 = - 3x - 4a
[Write it as disjunction.]

- 2x = 2 + 4a or 4x = 2 - 4a

x = - 1 - 2a or x = 12 - a

Check the answer: x = - 1 - 2a

|- 1 - 2a - 2| = 3 (- 1 - 2a) + 4a

|- (3 + 2a)| = - 3 - 6a + 4a

3 + 2a = - 3 - 2a, which is not true.
[Use |- x| = |x| = x, x is a positive real number.]

Hence, x = - 1 - 2a is not the solution.

Check the answer: x = 12 - a

|12 - a - 2| = 3 (12 - a) + 4a

|(1 - 2a - 4)2| = 3((1 - 2a)2) + 4a

|(- 2a - 3)2| = 3 - 6a + 8a2

|- (2a + 3)2| = (3 + 2a)2

(2a + 3)2 = (2a + 3)2 , which is true.
[Use |- x| = |x| = x, x is a positive real number.]

So, x = 12 - a is the solution.

32.
What are the values of $x$, if |$x$ - 7| - 4 = - 0.14?
 a. 10.86, 2.14 b. 10.86, 3.14 c. 9.86, 3.14 d. 9.86, 2.14

#### Solution:

|x - 7| - 4 = - 0.14

|x - 7| - 4 + 4 = - 0.14 + 4
[Add 4 to both sides of the equation.]

|x - 7| = 3.86
[Simplify.]

If |x - 7| = 3.86, then x - 7 equals 3.86 or - 3.86.

x - 7 = 3.86 or x - 7 = - 3.86

x - 7 + 7 = 3.86 + 7 or x - 7 + 7 = - 3.86 + 7
[Add 7 to both sides of the equation.]

x = 10.86 or x = 3.14
[Simplify.]

The solutions of the equation are 10.86 and 3.14.

33.
Find the number of solutions for the equation. |$x$ - 2| + 1 = 5
 a. 3 b. 1 c. 2

#### Solution:

|x - 2| + 1 = 5

|x - 2| = 4
[Isolating the equation.]

x - 2 equals 4 or - 4.

x - 2 + 2 = 4 + 2 or x - 2 + 2 = - 4 + 2.
[Add 2 to both sides of the equation.]

x = 6 or x = -2
[Simplify.]

The values of x are 6 and -2.

So, the equation has two solutions.