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Which equation gives the values of all numbers seven units away from 4

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Which equation gives the values of all numbers seven units away from 4  [#permalink]

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New post 17 Apr 2018, 01:14
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Which equation gives the values of all numbers seven units away from 43?

A. |x + 7| = 43
B. |x – 7| = 43
C. |x – 43| = 14
D. |x – 43| = 7
E. |x + 43| = 7

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Re: Which equation gives the values of all numbers seven units away from 4  [#permalink]

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New post 17 Apr 2018, 01:43
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Answer is D by definition of absolute value function.
|x – a| = b gives set of points x which are at a distance of b unit from a.
Hence |x – 43| = 7 gives all value of x at a distance of 7 from 43.

Answer D
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Which equation gives the values of all numbers seven units away from 4  [#permalink]

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New post 17 Apr 2018, 18:57
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Bunuel wrote:
Which equation gives the values of all numbers seven units away from 43?

A. |x + 7| = 43
B. |x – 7| = 43
C. |x – 43| = 14
D. |x – 43| = 7
E. |x + 43| = 7

gmatbusters posted the fastest and easiest way to solve, IMO. If you cannot recall the formulation, or get mixed up, a couple of details might help.

1) watch the SIGN inside the brackets
|x| = the distance OF x from 0
|x - 3| is the distance OF x FROM +3
We measure distance by subtracting. There's one hint:

A minus sign inside the brackets indicates distance of x from a POSITIVE number.

The distance of x from a negative number is written
x - (-3), which = (x + 3)

In this context, distance from a negative number will be indicated by a plus sign (+) inside the brackets.

Even if you're confused about where 7 and 43 go, both are positive. Whatever is INSIDE the brackets will be preceded by a minus sign.
Eliminate answers A and E

2) Take the problem at face value. It mentions the numbers 43 and 7, not 14. Ignore C for now

We're down to answers B and D. Solve them. Drawing a number line helps here, I think.

B. |x – 7| = 43
Case 1: x - 7 = 43
x = 50
Case 2: -(x - 7) = 43
-x + 7 = 43
-x = 36
x = -36

On this number line, boldface type = a discrete dot. There are two points, x = -36 and x = 50. Nothing in between is highlighted.

<--(-36)-------(0)---------------(50)-->

Prompt says "all numbers [that are] seven units away from 43." Write in 43 on the number line.

<--(-36)-----(0)--------(43)--(50)-->
Well, 50 is indeed 7 units away from 43.

But -36? No, -36 is many more than seven units away from 43. WRONG.
No need to calculate. It is obvious that -36 is more than seven units away from 43.
Then again, this can be a tough crowd... the distance of -36 from 43 is: 43 - (-36) = 79

Eliminate B

D. |x – 43| = 7
Case 1: x - 43 = 7
x = 50
Case 2, -(x - 43) = 7
-x + 43 = 7
-x = -36)
x = 36

At this point the logic might be clearer, such that you could investigate 36 and 50 in relation to 43. If not, draw a number line.

<--(0)------(36)----------------(50)-->

Back to the prompt: "all numbers that are seven units away from 43." Write in 43

<--(0)------(36)-----(43)------(50)-->

36 is seven units away from 43
50 is seven units away from 43
We have gone both directions on the number line.
There are no other numbers that are exactly 7 units away from 43. CORRECT

Answer D

For absolute value as distance, an exceptional introduction is mikemcgarry ,GMAT Math: Understanding Absolute Values

Other superb resources include but are not limited to chetan2u , Absolute Modulus, a Better Understanding
and Bunuel , Ultimate GMAT Quantitative Megathread: 10. Absolute Value
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Re: Which equation gives the values of all numbers seven units away from 4  [#permalink]

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New post 19 Apr 2018, 17:55
Bunuel wrote:
Which equation gives the values of all numbers seven units away from 43?

A. |x + 7| = 43
B. |x – 7| = 43
C. |x – 43| = 14
D. |x – 43| = 7
E. |x + 43| = 7


If x represents all the values that are a units from a number k, then |x - k| = a. Thus, we see that a = 7 and k = 43, so here we have:

|x - 43| = 7

Alternate Solution:

First, let’s note that there are only two numbers that are 7 units away from 43, and they are (43 – 7 ) = 36 and (43 + 7) = 50. Thus, we are looking for the solution that gives 36 and 50 as the only answers for x. Let’s test each answer choice:

Answer Choice A:

Case 1: (x + 7) is positive

x + 7 = 43

x = 36 This works.

Case 2: (x + 7) is negative

-(x + 7) = 43

-x – 7 = 43

-x = 36

x = -36 This doesn’t work, so we eliminate choice A

Answer Choice B:

Case 1: (x - 7) is positive

x – 7 = 43

x = 50 This works

Case 2: (x – 7) is negative

-(x – 7) = 43

-x + 7 = 43

-x = 36

x = -36 This doesn’t work. Eliminate Choice B

Answer choice C:

Case 1: (x – 43) is positive

x – 43 = 14

x = 57 This doesn’t work. Eliminate Choice C.

Answer Choice D:

Case 1: (x – 43) is positive

x – 43 = 7

x = 50 This works.

Case 2: (x – 43) is negative

-(x – 43) = 7

-x + 43 = 7

-x = -36

X = 36 This works! Answer Choice D is correct.

Answer: D
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Re: Which equation gives the values of all numbers seven units away from 4 &nbs [#permalink] 19 Apr 2018, 17:55
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