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Absolute Value Function Worksheet - Page 4

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


Correct answer : (4)
 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.


Correct answer : (2)
 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.


Correct answer : (4)

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