GMATPrepNow wrote:

Bunuel 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

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 = -k2. 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 16Range =

16 -

8 = 8

Answer:

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 = 0Factor: (x - 10)(x - 8) = 0

So, x = 10 or x = 8

When you do -56 - 24 why did you get a positive 80 and not negative 80?