If x≠4, what is the range of the solutions of the equation |14

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Re: If x≠4, what is the range of the solutions of the equation |14–x|=24/( [#permalink]

13 Jul 2017, 18:40

Bunuel wrote:

If x≠4, what is the range of the solutions of the equation $|14–x|=\frac{24}{(x−4)}$?

A. 2
B. 6
C. 8
D. 20
E. 32

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.

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Re: If x≠4, what is the range of the solutions of the equation |14–x|=24/( [#permalink]

16 Jul 2018, 10:14

Bunuel wrote:

If x≠4, what is the range of the solutions of the equation $|14–x|=\frac{24}{(x−4)}$?

A. 2
B. 6
C. 8
D. 20
E. 32

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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Re: If x≠4, what is the range of the solutions of the equation |14–x|=24/( [#permalink]

30 Oct 2018, 06:50

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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 = -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:

Show Spoiler

C

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?

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Re: If x≠4, what is the range of the solutions of the equation |14–x|=24/( [#permalink]

27 Apr 2021, 05:59

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.

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Re: If x≠4, what is the range of the solutions of the equation |14–x|=24/( [#permalink]

27 Apr 2021, 22:01

Bambi2021 wrote:

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.

D

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If x≠4, what is the range of the solutions of the equation |14–x|=24/( [#permalink]

24 Dec 2021, 09:53

Expert Reply

Top Contributor

Given that x≠4 and we need to find the range of the solutions of the equation $|14–x|=\frac{24}{(x−4)}$

Now, in order to solve this equation we need to deal with the Absolute Value of |14-x|
We can do that by taking two cases

Case 1: 14 -x >= 0 or x <= 14
=> | 14-x | = 14 -x
=> $14–x = \frac{24}{(x−4)}$
=> (14-x) * ( x-4) = 24
We can either use Algebra (open the equation and solve) or can substitute values

Substitution

(14-x) * ( x-4) = 4 * 6
If we put x = 10 then we will get
(14-10) * (10-4) = 4*6
So, x = 10

(14-x) * ( x-4) = 6 * 4
If we put x = 8 then we will get
(14-8) * (8-4) = 6*4
So, x = 8

Algebra

(14-x) * ( x-4) = 24
=> 14x - 56 - $x^2$ + 4x = 24
=> - $x^2$ + 18x - 56 - 24 = 0
=> - $x^2$ + 18x - 80 = 0
Multiply by -1 we get
=> $x^2$ - 18x + 80 = 0
=> $x^2$ - 8x - 10x + 80 = 0
=> x*(x-8) - 10*(x-8) = 0
=> (x-8) * (x-10) = 0
=> x = 8 or 10

Both values are <= 14
=> x can be 8 or 10

Case 2: 14 -x < 0 or x > 14
=> | 14-x | = -(14 -x) = x - 14
=> $x-14 = \frac{24}{(x−4)}$
=> (x-14) * ( x-4) = 24
We can either use Algebra (open the equation and solve) or can substitute values

Substitution

(x-14) * (x-4) = 2 * 12
If we put x = 16 then we will get
(16-14) * (16-4) = 2*12
So, x = 16
Since 16 > 14
=> x = 16 is a solution

Algebra

(x-14) * ( x-4) = 24
=> $x^2$ - 4x -14x + 56 = 24
=> $x^2$ - 18x + 56 - 24 = 0
=> $x^2$ - 18x + 32 = 0
=> $x^2$ - 2x - 16x + 32 = 0
=> x*(x-2) - 16*(x-2) = 0
=> (x-2) * (x-16) = 0
=> x = 2 or 16
But x > 14
=> x = 16 is a solution

Combining both the cases possible values of x are 8, 10, 16

Range = Highest value - Lowest value = 16-8 = 8

So, Answer will be C
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

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If x4, what is the range of the solutions of the equation |14x|=24/( [#permalink]

04 Feb 2024, 21:47

Quote:

BrentGMATPrepNow

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 = -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:

Show Spoiler

C

Hi @KarishmaB,

In the 2nd case - Multiply both sides by (x-4) to get: (14–x)(x−4) = -24, upon solving the equation we received the values 16 and 2. However these are not negative values as per our assumption for the expression. Hence, shouldn't they be outright rejected instead of testing?

I am trying to understand the testing rationale here.

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Re: If x4, what is the range of the solutions of the equation |14x|=24/( [#permalink]

05 Feb 2024, 00:09

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Expert Reply

Bunuel wrote:

If x≠4, what is the range of the solutions of the equation $|14–x|=\frac{24}{(x−4)}$?

A. 2
B. 6
C. 8
D. 20
E. 32

When we have a mix and x and |x| in an equation, there is only one thing to do - Take cases.

$|14–x|=\frac{24}{(x−4)}$
is the same as
$|x-14|=\frac{24}{(x−4)}$

Two cases:
Case 1: x >= 14
$x-14=\frac{24}{(x−4)}$
$x^2 -18x + 32 = 0$
x = 2 or 16
Only 16 is greater than 14 hence only 16 is an acceptable answer.

Case 2: x< 14
$x-14=\frac{-24}{(x−4)}$
x^2 - 18x + 80 = 0
x = 8 or 10
Both are acceptable since both are less than 14

Range of solutions = Greatest solution - Least solution = 16 - 8 = 8

Answer (C)

Check this post: https://anaprep.com/algebra-the-why-beh ... questions/

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KarishmaB

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Re: If x4, what is the range of the solutions of the equation |14x|=24/( [#permalink]

05 Feb 2024, 00:14

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Expert Reply

MK010 wrote:

Quote:

Hi @KarishmaB,

In the 2nd case - Multiply both sides by (x-4) to get: (14–x)(x−4) = -24, upon solving the equation we received the values 16 and 2. However these are not negative values as per our assumption for the expression. Hence, shouldn't they be outright rejected instead of testing?

I am trying to understand the testing rationale here.

BrentGMATPrepNow

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 = -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:

Show Spoiler

C

MK010
First, simplify |14 - x| as |x - 14| since both are the same.
After that, take two cases as I have shown above. Once you get your solutions, all you need to do is check the original condition and accept solutions as per that. The ones that do not fall under our original condition should be outright rejected. No testing needed.

gmatclubot

Re: If x4, what is the range of the solutions of the equation |14x|=24/( [#permalink]

05 Feb 2024, 00:14

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