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Absolute value & the number line

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BlueDragon2010 Senior | Next Rank: 100 Posts Default Avatar
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Absolute value & the number line

Post Wed Feb 19, 2014 7:40 am


Which of the following inequalities is an algebraic expression for the shaded part of the number line above?

A) |x| <= 3
B) |x| <= 5
C) |x-2| <= 3
D) |x-1| <= 4
E) |x+1| <= 4

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Patrick_GMATFix GMAT Instructor
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Post Wed Feb 19, 2014 7:52 am
One approach: plug in the two boundaries from the graph (-5 and 3) into each answer choice. In the wrong answers, at least one of the numbers will result in an invalid statement. In the right answer, both of these numbers will result in a valid equality. For instance plugging -5 and 3 into |x+1|<=4 will result in 4=4. This is the right answer.

The full solution below is taken from the GMATFix App.



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Post Wed Feb 19, 2014 7:59 am
BlueDragon2010 wrote:


Which of the following inequalities is an algebraic expression for the shaded part of the number line above?

A) |x| <= 3
B) |x| <= 5
C) |x-2| <= 3
D) |x-1| <= 4
E) |x+1| <= 4
|x-y| = the DISTANCE between x and y

The midpoint of the range = -1.
x can be any value from -5 (4 places BELOW -1) to 3 (4 places ABOVE -1).
In other words: the distance between x and -1 is less than or equal to 4.
In math terms:
|x-(-1)| ≤ 4.
|x+1| ≤ 4.

The correct answer is E.

An alternate approach is to plug numbers into the answer choices.
The correct answer choice must work for EVERY value between -5 and 3, inclusive -- and for no values outside this range.
Since |x| = the distance between x and 0, start with the value furthest from 0:
Let x=-5.
Eliminate any answer choice in which x=-5 doesn't work.

Answer choice A: |x|≤3
|-5|≤3.
5≤3.
Doesn't work. Eliminate A.

Answer choice B: |x|≤5
|-5|≤5
5≤5.
This works. Hold onto B.

Answer choice C: |x-2|≤3
|-5-2|≤3
7≤3.
Doesn't work. Eliminate C.

Answer choice D: |x-1|≤4
|-5-1|≤4
6≤4.
Doesn't work. Eliminate D.

Answer choice E: |x+1|≤4
|-5+1|≤4
4≤4.
This works. Hold onto E.

Only B and E remain.
Answer choice B -- |x|≤5 -- includes every value from -5 to 5, but the needed range is only from -5 to 3.
Eliminate B.

The correct answer is E.

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Post Wed Feb 19, 2014 1:07 pm
Hi BlueDragon2010,

Patrick and Mitch have both pointed out how useful the answer choices can be to solving this question. The GMAT writers create questions (for both the Quant and Verbal sections) that can usually be answered in a variety of ways. Many of those questions come with "hints" as to which option is fastest. In this case, the five answers are equations so one of them MUST be the equation that describes what you see in the number line. Using those 5 options as a gauge, you can quickly deduce which is a match and which 4 are not. You'll see several questions on Test Day in which TESTING THE ANSWERS will be the fastest option to solving the problem, so keep this approach in mind as you study.

GMAT assassins aren't born, they're made,
Rich

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