ktrout2020 wrote:If x ≠4, what is the range of the solutions of the equation |14 - x| = 24/(x − 4) ?
A. 2
B. 6
C. 8
D. 20
E. 32
Source: Veritas
There are 3 steps to solving equations involving ABSOLUTE VALUE:
1. Apply the rule that says:
If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug solutions into original equation to check for extraneous roots
Given: |14-x|=24/(x−4)
So, we need to check
14-x = 24/(x−4) and
14-x = -24/(x−4)
14-x = 24/(x−4)
Multiply both sides by (x-4) to get: (14-x)(x−4) = 24
Expand: -x² + 18x - 56 = 24
Rearrange to get: x² - 18x + 80 = 0
Factor: (x - 10)(x - 8) = 0
So, x = 10 or x = 8
Test each solution:
If x = 10, then we get: |14-10|=24/(10−4)
Simplify: |4|=4 PERFECT!
So,
x = 10 is a possible solution
If x = 8, then we get: |14-8|=24/(8−4)
Simplify: |6|=6 PERFECT!
So,
x = 8 is a possible solution
14-x = -24/(x−4)
Multiply both sides by (x-4) to get: (14-x)(x−4) = -24
Expand: -x² + 18x - 56 = -24
Rearrange to get: x² - 18x + 32 = 0
Factor: (x - 2)(x - 16) = 0
So, x = 2 or x = 16
Test each solution:
If x = 2, then we get: |14-2|=24/(2−4)
Simplify: |12|=-12 DOESN'T WORK
So, x = 2 is NOT a possible solution
If x = 16, then we get: |14-16|=24/(16−4)
Simplify: |-2|=2 PERFECT!
So,
x = 16 is a possible solution
So, the possible solutions are
8, 10 and 16
Range =
16 -
8 = 8
Answer: C
Cheers,
Brent
