Solve for x: 0<|x|-4x<5 = ?

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vinni.k: Master | Next Rank: 500 Posts; Posts: 421; Joined: Sun Apr 17, 2011 4:27 amFollowed by: 2 members; Location: India; Thanked: 6 times; GMAT Score:620

by vinni.k » Mon Jan 13, 2014 8:44 am

Solve for x: 0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

OA is E

I am solving this problem in the following algebraic way and not by number plugging as it is confusing me.

There are two cases
(1). 0 < -3x < 5
(2). 0 < -5x < 5

From here how should i proceed.

Thanks & Regards
Vinni

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Source: Beat The GMAT — Problem Solving | Mon Jan 13, 2014 8:44 am

theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Mon Jan 13, 2014 9:30 am

0<|x|-4x<5

Case 1:
0 < x-4x < 5
0 < -3x < 5
0 < -x < 5/3
==> x < 0 & x > -5/3

Case 2:
0 < -x-4x < 5
0 < -5x < 5
0 < -x < 1
==> x < 0 & x > -1

If we compare the answer choices, we see that [spoiler]{E}[/spoiler] covers maximum values.

R A H U L

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Jan 13, 2014 9:46 am

vinni.k wrote:Solve for x: 0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

OA is E

A: x<0
Test x=-10 in 0<|x|-4x<5:
0 < |-10| -4(-10) < 5
0 < 50 < 5.
Doesn't work.
Eliminate A.

B: 0 < x < 1
Test x=1/2 in 0<|x|-4x<5:
0 < |1/2| - 4(1/2) < 5
0 < -1.5 < 5.
Doesn't work.
Eliminate B and C, each of which allows x=1/2.

D. -3/5<x<0
E. -1<x<0
Test x = -3/4 -- a value BETWEEN -1 and -3/5 -- in 0<|x|-4x<5:
0 < |-3/4| - 4(-3/4) < 5
0 < 2.25 < 5.
This works.
Eliminate D, which does not allow x=-3/4.

The correct answer is E.

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vinni.k: Master | Next Rank: 500 Posts; Posts: 421; Joined: Sun Apr 17, 2011 4:27 amFollowed by: 2 members; Location: India; Thanked: 6 times; GMAT Score:620

by vinni.k » Tue Jan 14, 2014 8:14 am

GMATGuruNY wrote:Solve for x: 0<|x|-4x<5 = ?
D. -3/5<x<0
E. -1<x<0
Test x = -3/4 -- a value BETWEEN -1 and -3/5 -- in 0<|x|-4x<5:
0 < |-3/4| - 4(-3/4) < 5
0 < 2.25 < 5.
This works.
Eliminate D, which does not allow x=-3/4.

The correct answer is E.

Mitch this is the exact point where i am getting confused and not solving by numbers..
Why do we go with x = -3/4 ? Why can't we just stick to D's inequality.
I took two numbers -2/5, and -1/5 and was getting 0<2<5 and 0<1<5 and same for E.

not sure how to follow.

Vinni

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Jan 14, 2014 1:08 pm

vinni.k wrote:
GMATGuruNY wrote:Solve for x: 0<|x|-4x<5 = ?
D. -3/5<x<0
E. -1<x<0
Test x = -3/4 -- a value BETWEEN -1 and -3/5 -- in 0<|x|-4x<5:
0 < |-3/4| - 4(-3/4) < 5
0 < 2.25 < 5.
This works.
Eliminate D, which does not allow x=-3/4.

The correct answer is E.
Mitch this is the exact point where i am getting confused and not solving by numbers..
Why do we go with x = -3/4 ? Why can't we just stick to D's inequality.
I took two numbers -2/5, and -1/5 and was getting 0<2<5 and 0<1<5 and same for E.

not sure how to follow.

Vinni

D. -3/5<x<0
E. -1<x<0

Strategy:
Test a value that is included in one range (-1<x<0) but NOT in the other range (-3/5<x<0).
x = -3/4 is included E's range but not in D's range.
When we test x=-3/4, we discover that it satisfies 0<|x|-4x<5:
0 < |-3/4| - 4(-3/4) < 5
0 < 2.25 < 5.
Thus, the correct range must include x=-3/4.
Since the range in D does not include x=-3/4, eliminate D.

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vinni.k: Master | Next Rank: 500 Posts; Posts: 421; Joined: Sun Apr 17, 2011 4:27 amFollowed by: 2 members; Location: India; Thanked: 6 times; GMAT Score:620

by vinni.k » Tue Jan 14, 2014 8:13 pm

GMATGuruNY wrote: D. -3/5<x<0
E. -1<x<0
Strategy:
Test a value that is included in one range (-1<x<0) but NOT in the other range (-3/5<x<0).
x = -3/4 is included E's range but not in D's range.

Mitch, this strategy is good, but if we don't take values from D's range -3/5<x<0, then doesn't it mean that we are not using inequality of D ? I mean to say i take one value that is included in one range (-1<x<0) but NOT in the other range (-3/5<x<0), now it gives me an impression that i am not using choice D for checking. Something out of the range will definitely not satisfy the inequality.

OR

What i have understood from your reply is that one choice needs to be eliminated (as there is only one answer in PS) . If i include a value for instance -2/5 or -1/5, it satisfies both the inequalities, but with your strategy D can be eliminated. Hence, answer is E.

I hope i am moving in a right direction.

Regards
Vinni

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed Jan 15, 2014 12:00 pm

vinni.k wrote:
GMATGuruNY wrote: D. -3/5<x<0
E. -1<x<0
Strategy:
Test a value that is included in one range (-1<x<0) but NOT in the other range (-3/5<x<0).
x = -3/4 is included E's range but not in D's range.
Mitch, this strategy is good, but if we don't take values from D's range -3/5<x<0, then doesn't it mean that we are not using inequality of D ? I mean to say i take one value that is included in one range (-1<x<0) but NOT in the other range (-3/5<x<0), now it gives me an impression that i am not using choice D for checking. Something out of the range will definitely not satisfy the inequality.

The correct answer choice must include ALL POSSIBLE VALUES OF X.
In testing x=-3/5 -- a value within E's range but outside D's range -- we are in fact checking whether D includes ALL possible values of x.
Since x=-3/5 satisfies the inequality in the question stem, D's range does NOT include all possible values of x.
Eliminate D.

Another case:
If |2x - 3| < 11, which of the following expresses all possible values of x?
A: -7 < x < 4
B: -4 < x < 7

Here, we should test a value between the two upper boundaries (4 and 7) or between the two lower boundaries (-7 and -4).
Test x=5, which is included in B's range but not in A's range:
|2*5 - 3| < 11
7 < 11.
This works.
The correct answer choice must include x=5 within its range.
Eliminate A.

The correct answer is B.

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Wed Jan 15, 2014 12:07 pm

Just to expand a bit on Rahul's approach, which would be my preferred way to solve this problem:

Given an inequality like 2 < |x|, we have to consider two scenarios:

(1) x is nonnegative, in which case |x| = x
(2) x is negative, in which case |x| = -x

So 2 < |x| implies:

(1) 2 < x, if x is nonnegative
(2) 2 < -x if x is negative

Given our inequality, 0<|x|-4x<5, we have

(1) 0 < x - 4x < 5, or 0 < -3x < 5, or 0 > x > -(5/3), if x is nonnegative, and
(2) 0 < -x - 4x < 5, or 0 < -5x < 5, or 0 > 5x > -5, or 0 > x > -1, if x is negative.

Now let's examine the two inequalities together:

0 > x > -(5/3)
0 > x > -1

Since we need an inequality that works for ALL values of x, we take the more conservative bounds: the SMALLER of the two maximums (here 0 and 0, so that's easy) and the GREATER of the two minimums (here -(5/3) and -1, so we'll take -1).

That means that our inequality is 0 > x > -1, and we're set!

Once you get the hang of this, it is much quicker, cleaner, and more reliable than number picking.

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vinni.k: Master | Next Rank: 500 Posts; Posts: 421; Joined: Sun Apr 17, 2011 4:27 amFollowed by: 2 members; Location: India; Thanked: 6 times; GMAT Score:620

by vinni.k » Thu Jan 16, 2014 7:37 am

Matt and Mitch, thank you so much for your explanations. You guys are just "Great" and I really really appreciate for your responses.

I am good to go now. Thanks once again.

Regards
Vinni

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