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Manager  Joined: 02 Dec 2012
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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
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31 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.

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

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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) Originally posted by adeelahmad on 11 Jan 2014, 03:18.
Last edited by adeelahmad on 11 Jan 2014, 03:26, edited 1 time in total.
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Manager  Joined: 20 Dec 2013
Posts: 116
Re: Which of the following inequalities is an algebraic expressi  [#permalink]

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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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Intern  Joined: 14 Jul 2013
Posts: 22
Re: Which of the following inequalities is an algebraic expressi  [#permalink]

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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.
Intern  Joined: 03 Jan 2014
Posts: 2
WE: Information Technology (Computer Software)
Re: Which of the following inequalities is an algebraic expressi  [#permalink]

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1
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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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.

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

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1
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.
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Re: Which of the following inequalities is an algebraic expressi  [#permalink]

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1
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) 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 ..
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Joined: 04 Sep 2016
Posts: 1366
Location: India
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Which of the following inequalities is an algebraic expressi  [#permalink]

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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
Attachments OG16 Q168.jpeg [ 85.35 KiB | Viewed 17521 times ]

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Originally posted by adkikani on 27 Aug 2017, 03:01.
Last edited by adkikani on 27 Aug 2017, 03:39, edited 1 time in total.
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
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Bunuel mikemcgarry IanStewart shashankism Engr2012

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

WR,
Arpit

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

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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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Posts: 58445
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1
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

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

For more check Ultimate GMAT Quantitative Megathread

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

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Bunuel 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

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.

Hello Bunuel, I have two questions: Why in options C and D you add 1 to all parts, whereas in option E you subtract 1 from all parts ? And another question: if you subtract +1 from -4 and +4 how do you get -5 and 3 ?
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Re: Which of the following inequalities is an algebraic expressi  [#permalink]

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dave13, niks18

Quote:
I have two questions: Why in options C and D you add 1 to all parts, whereas in option E you subtract 1 from all parts ?

While dealing with inequality we always compare a variable x wrt numbers, not x-1 or x+1
Note that in C and D we have added +2 and +1 resp, but in E we have added -1 on both sides.

Quote:
And another question: if you subtract +1 from -4 and +4 how do you get -5 and 3 ?

Adding or subtracting same no from inequality does not change the inequality.
Multiplying -1 on both sides changes inequality.

Hope this helps!
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Re: Which of the following inequalities is an algebraic expressi  [#permalink]

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dave13 wrote:
Bunuel 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

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.

Hello Bunuel, I have two questions: Why in options C and D you add 1 to all parts, whereas in option E you subtract 1 from all parts ? And another question: if you subtract +1 from -4 and +4 how do you get -5 and 3 ?

1. We add or subtract to get only x in the middle.
2. -4 - 1 = -5 and 4 - 1 = 3.
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Which of the following inequalities is an algebraic expressi  [#permalink]

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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) 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 Retired Moderator V
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Re: Which of the following inequalities is an algebraic expressi  [#permalink]

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2

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

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

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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

Mid point = -1
Range = 8

|x+1|<=4

IMO E
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