BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem
TTP OnDemand GMAT Video Course
Featured Course

Target Test Prep GMAT OnDemand

Scott Woodbury-Stewart’s private virtual classroom — 400 hours of master-class video lessons for the GMAT Focus Edition.

  • 715+ score guarantee — highest in the industry (99th percentile)
  • 52 chapters · 1,500+ lessons · 4,000+ practice questions
  • 400 hours of video · 1,500+ instructor-led HD smartboard lessons
  • 300,000+ students accepted to Harvard, Stanford, Wharton, Booth & Sloan & more
  • 24/7 live support + weekly Zoom office hours with GMAT instructors
  • TTP AI Assist — 24/7 AI-powered virtual tutor for instant help
  • 1,200+ flashcards + AI-powered study assistant & daily calendar
  • OnDemand, LiveTeach & GMAT Bootcamp formats available
  • Also: GRE, SAT Math & Executive Assessment courses
  • MBA Admissions Consulting now available
  • 🏆 2025 EdTech Breakthrough Award: Test Prep Solution Provider of the Year
  • 200,000+ students served
  • 5-day free trial — $0 to start, no auto-billing, cancel anytime
★★★★★ 5.0 (559 reviews)
130-pt guarantee
$0 to start then $127/mo
Start Free 5-Day Trial Read 559 Reviews →

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

Expert replies
Post new topic Post Reply
Previous Topic Next Topic
Source: — Problem Solving |

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3008
Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members
Location: Grand Central / New York
Thanked: 470 times

by Jay@ManhattanReview » Thu Nov 07, 2019 11:32 pm
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
_________________
Manhattan Review

Locations: Manhattan Review Mumbai | GMAT Prep New Delhi | GRE Prep Malleswaram | Vijayawada GRE Coaching | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members
Location: Vancouver, BC
Thanked: 5254 times
GMAT Score:770

by Brent@GMATPrepNow » Fri Nov 08, 2019 6:20 am
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
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 8147
Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members
Location: Los Angeles, CA
Thanked: 43 times

by Scott@TargetTestPrep » Tue Nov 19, 2019 6:14 pm
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

Scott Woodbury-Stewart
Founder and CEO
[email protected]

Image

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

ImageImage
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Post new topic Post Reply
• Page 1 of 1