If x≠4, what is the range of the solutions of the equation

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

We are given that |14 – x| = 24/(x − 4)

Let's take |14 – x| as 14 – x

Thus, we have 14 – x = 24/(x − 4)

=> x^2 – 18x + 80 = 0
x^2 – 10x – 8x + 80 = 0
(x – 8)(x – 10) = 0

So, x = 8 or 10

Now let's take |14 – x| as –14 + x

Thus, we have –14 + x = 24/(x − 4)

=> x^2 – 18x + 32 = 0
x^2 – 16x – 2x + 80 = 0
(x – 16)(x – 2) = 0

So, x = 2 or 16

Note that @x = 2, we have |14 - x| = 24/(x - 4) => |14 - 2| = 24/(2 - 4) => 12 ≠ – 12; thus x = 2 is not a solution.

Thus, the three solutions are 8,10 and 16. Thus, the range = 16 – 8 = 8

The correct answer: C

Hope this helps!

-Jay
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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

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

Case 1. When (14 - x) is positive, we have:

14 - x = 24/(x - 4)

(14 - x)(x - 4) = 24

-x^2 + 18x - 56 = 24

x^2 - 18x + 80 = 0

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

x = 8 or x = 10

If we check these two solutions in the original equation, they both satisfy the equation.

Case 2. When (14 - x) is negative, we have:

-14 + x = 24/(x - 4)

(-14 + x)(x - 4) = 24

x^2 - 18x + 56 = 24

x^2 - 18x + 32 = 0

(x - 16)(x - 2) = 0

x = 16 or x = 2

If we check these two solutions in the original equation, only 16 satisfies the equation. Therefore, only 16 is the solution when (14 - x) is negative.

Thus, the range of the solutions is 16 - 8 = 8.

Answer: C