Which of the following inequalities is an algebraic expression for the

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

From the number line it follows that \(-5\leq{x}\leq{3}\)


(A) |x| <= 3 --> \(-3\leq{x}\leq{3}\). Discard.

(B) |x| <= 5 --> \(-5\leq{x}\leq{5}\). Discard.

(C) |x - 2| <= 3 --> \(-3\leq{x-2}\leq{3}\) --> add 2 to all parts: \(-1\leq{x}\leq{5}\). Discard.

(D) |x - 1| <= 4 --> \(-4\leq{x-1}\leq{4}\) --> add 1 to all parts: \(-3\leq{x}\leq{5}\). Discard.

(E) |x +1| <= 4 --> \(-4\leq{x+1}\leq{4}\) --> subtract 1 from all parts: \(-5\leq{x}\leq{3}\). OK.


Answer: E.
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Lets try to do this conceptually,

The length of the line is 8. Middle point = 8/2 = 4. The point on the number line equidistant at a length of 4 from each extremeties (-5 and 3) is -1. So, the equation turns out to be,

|x - (equidistant point)| <= Middle Point
i.e. |x-(-1)| <= 4
i.e. |x+1| <= 4

Ans - (E)

Round 1: Eliminate the obvious, let us say x = -5 and eliminate from options

(A) Eliminated
(B) Okay
(C) Eliminated
(D) Eliminated
(E) Okay

Round 2: We are left between B and E. There are two things you can do.

Use Algebra:
|x| <= 5 - Contains numbers from -5 to +5 which does not define the inequality as 4 and 5 are not part of the inequality

Hence answer is E

OR

Plug in x = 4 where Option B satisfies which was not supposed to be.

My Approach

Looking at the number line, it can be inferred that the line would have been \(|x|\leq{4}\) or -\(4\leq{x}\leq{4}\) if it was was centered at 0. (since length = 8 units and end points as +-4)

Now since the line \(|x|\leq{4}\) is centered at 0 and in this case is shifted to left by 1 unit (now centered at -1), the equation of the line becomes \(|x-(-1)|\leq{4}\)
or \(|x+1|\leq{4}\)

Hence E

We start by expressing the interval on the number line as an inequality:

-5 ≤ x ≤ 3

Looking at answer choices A and B, we see that those two equations will not produce the inequality shown above. Thus, we consider answer choices C, D, and E.

When we solve an absolute-value equation with one absolute-value expression, we consider two cases: one with the positive version of the expression inside the absolute value bars and one with the negative (or opposite) version of the expression inside the absolute value bars. Let’s use this fact to evaluate answer choice C:

Answer choice C: |x - 2| ≤ 3

Case 1: Expression Positive:

x – 2 ≤ 3

x ≤ 5

Case 2: Expression Negative:

-(x - 2) ≤ 3

-x + 2 ≤ 3

-x ≤ 1

x ≥ -1

The solution is x ≤ 5 and x ≥ -1, i.e., -1 ≤ x ≤ 5. However, this does not fit the interval represented on the number line.

Answer choice D: |x - 1| ≤ 4

Case 1: Expression Positive:

x - 1 ≤ 4

x ≤ 5

Case 2: Expression Negative:

-(x – 1) ≤ 4

-x + 1 ≤ 4

-x ≤ 3

x ≥ -3

The solution is x ≤ 5 and x ≥ -3, i.e., -3 ≤ x ≤ 5. This does not fit the interval represented on the number line.

Answer Choice E: |x +1| ≤ 4

Case 1: Expression Positive:

x + 1 ≤ 4

x ≤ 3

Case 2: Expression Negative:

-(x + 1) ≤ 4

-x – 1 ≤ 4

-x ≤ 5

x ≥ -5

The solution is x ≤ 3 and x ≥ -5, i.e., -5 ≤ x ≤ 3. This DOES describe the interval represented on the number line.

Answer E
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Got this quickly using the Kaplan trick, start at the bottom for "which of the following" type questions. Didn't waste time evaluating the first four wrong ones.

beautiful approach I will say ...

can you please, however, say to me what you will do when you cannot find out the midpoint, specifically , in the cases when the length of the line is an odd number, for instance, suppose, 9....?

thanks in advance ..

Bunuel mikemcgarry IanStewart shashankism Engr2012

Can someone please explain statement (C) in Bunuel's approach in detail.
I got through OA by opening modulus, but took more steps and time.

WR,
Arpit

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For C you have the following mistake:

\(|x - 2| \leq 3\):

\(x - 2 \leq 3\) --> \(x \leq 5\)
\(-x + 2 \leq 3\) --> \(-x \leq 1\) --> \(x \geq -1\) NOT \(x \geq 1\) as you've written.
So, \(-1 \leq x \leq 5\)

You can check posts above for complete solution.

Hope it helps.
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hi Bunuel

Thanks a lot for pointing out my mistake.
What I meant earlier was explanation of point (C) in your approach since it was too concise
particularly this step:
(C) |x - 2| <= 3 --> −3≤x−2≤3−3≤x−2≤3 --> add 2 to all parts: −1≤x≤5−1≤x≤5. Discard.. Discard.

WR,
Arpit
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It's basically the same method as your but done in one line. You can check different approaches strategies in the posts below:

10. Absolute Value

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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OR

as can be seen, and it is very important to notice that "3" and "-5" both numbers are exactly "1" distance away from the middle point "4" to forming an algebraic expression
so the actual expression should be

-5 + 1 <= x + 1 <= 3 + 1
thus,

-4 <= x + 1 <= 4

thanks

Steps to convert inequality graph to algebraic expression:

1) Plot the midpoint (a) of the solution set on the number line:
here the midpoint, a = (-5+3)/2 = -1

2) Find the distance (b unit) of either end points from the mid point :
here the distance, b = 3-(-1) = 4

3) Insert the appropriate sign of inequality between |x-a| and b:
Here , putting a and b, we get |x+1|<=4

hence, the Required inequality is |x+1|<=4

Answer = E
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Bunuel, IanStewart, EMPOWERgmatRichC, ArvindCrackVerbal, AaronPond, VeritasKarishma, GMATinsight, ccooley

Hello Honorable Experts,
Could you help me to cancel choice B by testing number, please?
Thanks__
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Asad

The figure shows that x can have all values from -5 to +3 (both included)

B option when simplified gives value range -5 ≤ x ≤ +5

i.e. Value +5, which satisfies the Option B, doesn't satisfy the given inequality in graph (which ranges from -5 to +3) hence can be eliminated

Option A can be eliminated because it doesn't include value of x=-5 which the graph includes hence eliminated

Option C can be eliminated because it doesn't satisfy for value of x=-5 which the graph includes hence eliminated

Option D can be eliminated because it doesn't satisfy for value of x=-5 which the graph includes hence eliminated

Option E satisfies each value of x from -5 to +3 both included hence CORRECT OPTION

Answer: Option E

Hi Asad,

Based on how this question is written, we need an inequality that matches up COMPLETELY with the shaded section of the Number Line in the included picture. In simple terms, this means that the correct answer will NOT include any numbers that are not also included in that section of the Number Line.

With Answer B, both X=4 and X=5 would be potential values for X, but neither of those numbers is within the range defined by the Number Line, so this cannot be the Answer.

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

Hello Asad,

In my opinion, number testing is not needed to eliminate option B in this question. However, siince you wanted me to help you cancel choice B by testing numbers, I’ll do exactly that before I discuss what would be my preferred method of solving this question.

The interval shown in the diagram is -5≤x≤3.

Answer option B says |x| ≤ 5. Can we plug in positive 5 here to satisfy this inequality? We can. But, 5 cannot be a value for x since -5≤x≤3. Therefore, |x| ≤ 5 does not represent the algebraic expression for the shared part of the line given.

If |x| ≤ a, then -a≤ x ≤a. Based on this concept, we can eliminate answer option A and B since they do not represent the algebraic expression for the shaded part of the line.

|x-a| represents the distance of x from a. If |x-a| ≤ k, then a-k ≤ x ≤ a + k.

Answer option C: |x-2| ≤ 3. a = 2 and k = 3. Therefore, -1 ≤ x ≤ 3. Answer option C can be eliminated.

Answer option D: |x-1| ≤ 4. a = 1 and k = 4. Therefore, -3 ≤ x ≤ 5. Answer option D can be eliminated.

Answer option E HAS TO be the correct answer.
Answer option E: |x+1|≤ 4. a = -1 and k = 4. Therefore, -5 ≤ x ≤ 3 which is the algebraic expression for the shaded part.

Hope that helps!
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Though you have got enough replies on your query so I will not talk about this particular question, I would like to point out this:

"Testing numbers" is a mechanical strategy, not something that involves your mind much. So you will NOT get rewarded well for using it. A question that you can efficiently solve using testing numbers will be a very low level question.
It can help you eliminate some options, make it easier for you to see what the question is asking etc but it will not lead to the answer in a smarter question until and unless you use your ingenuity along with it. Do not blindly rely on testing numbers - try to put in more mental effort than that.
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