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:

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

(x - 4)(14 - x) = 24
x^2 - 18x + 80 = 0
x = 8 or 10

(x - 4)(14 - x) = -24
x^2 - 18x + 32 = 0
x = 16 or 2 but 2 is not a valid solution since \(|14 - 2| \neq {-12}\)

Range = 16 - 8 = 8

Answer: C

pushpitkc , Is it possible to solve this question using the number line?

It took me more than 2 minutes to solve this question. I normally prefer the number line, but in this case I just couldn't use it.

Given: \(|14–x|=\frac{24}{(x−4)}\)

For 14 - x > 0, we have x < 14

(14-x)(x-4) = 24, solving we get x = 10 or 8 < 14, hence both solutions are good.

For 14 - x < 0, we have x > 14

(x-14)(x-4) = 24. solving we get x = 2 or 16, only x = 16 > 14 is good.

hence we have x = 8, 10, 16

Range of Solutions = 16 - 8 = 8


Answer C.


Thanks,
GyM
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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:



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


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

I solved it the following way, but maybe it was just a coincidence that my approach worked in this case:

When we arrive at (x-4)(14-x) = 24, I just thought that the range must be between the values of x when any of the factors is 0.

x = 4 -> 0
x = 14 -> 0

Range is 5-13 = 8.

Guess I was just lucky this time.

Interesting point, but I don't think we are told that X is an integer. Just that when we solve, the max value is 16 and the minimum value is 8.