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Which of the following inequalities is an algebraic expressi

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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
[Reveal] Spoiler: OA
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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.. Discard.
(D) |x - 1| <= 4 --> \(-4\leq{x-1}\leq{4}\) --> add 1 to all parts: \(-3\leq{x}\leq{5}\). Discard.. 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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Re: Which of the following inequalities is an algebraic expressi [#permalink]

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Walkabout wrote:
Attachment:
Line.png
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



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) :-D

Last edited by adeelahmad on 11 Jan 2014, 03:26, edited 1 time in total.
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Re: Which of the following inequalities is an algebraic expressi [#permalink]

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New post 11 Jan 2014, 03:24
Walkabout wrote:
Attachment:
Line.png
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


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.
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Re: Which of the following inequalities is an algebraic expressi [#permalink]

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New post 21 Apr 2014, 04:29
yeehaaahhh

tried with bunuel's logic. It worked within 30 sec

calculate length - 8
center - from the graph = -1

only two options have r.h.s = 4 (half of length). Further, in E, LHS becomes zero when x = -1.
Hence, E.
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Re: Which of the following inequalities is an algebraic expressi [#permalink]

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New post 16 Jun 2014, 00:48
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
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Re: Which of the following inequalities is an algebraic expressi [#permalink]

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New post 16 Jun 2016, 06:18
Walkabout wrote:
Attachment:
Line.png
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


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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Re: Which of the following inequalities is an algebraic expressi   [#permalink] 16 Jun 2016, 06:18
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