√3-2|+|3-√2|+|5+√3|+|1-√2|=?

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[Math Revolution GMAT math practice question]

|√3-2|+|3-√2|+|5+√3|+|1-√2|=?

A. 0
B. 2√2
C. 2√3
D. 9
E. 11

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by GMATGuruNY » Wed Sep 19, 2018 2:40 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

|√3-2|+|3-√2|+|5+√3|+|1-√2|=?

A. 0
B. 2√2
C. 2√3
D. 9
E. 11
|a-b| = the distance between a and b.
|a+b| = |a-(-b| = the distance between a and -b.

|√3-2| + |3-√2| + |5+√3| + |1-√2|

= |5+√3| + |√3-2| + |1-√2| + |3-√2|

= |√3+5| + |√3-2| + |1-√2| + |3-√2|

= |√3-(-5)| + |√3-2| + |1-√2| + |3-√2|

The red terms constitute the sum of the following two distances:
-5<----->√3<----->2
The sum of these two distances = the distance between -5 and 2 = 7.

The blue terms constitute the sum of the following two distances:
1<----->√2<----->3
The sum of these two distances = the distance between 1 and 3 = 2.

Thus:
Sum of all 4 terms = 7+2 = 9.

The correct answer is D.
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by Max@Math Revolution » Fri Sep 21, 2018 12:17 am
=>

|A|=A when A>0, |0|=0, and |A|=-A when A<0

Since √3-2 < 0, we have |√3-2| = -(√3-2).
Since 3-√2 > 0, we have |3-√2| = 3-√2.
Since 5+√3 > 0, we have |5+√3| = 5+√3
Since 1-√2 < 0, we have |1-√2| = -(1-√2)
So, |√3-2|+|3-√2|+|5+√3|+|1-√2|= -(√3-2) +(3-√2) + (5+√3) -(1-√2)
= -√3+2 + 3-√2 + 5+√3 - 1 +√2 = 9.

Therefore, the answer is D.
Answer: D

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by Scott@TargetTestPrep » Thu Sep 27, 2018 4:28 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

|√3-2|+|3-√2|+|5+√3|+|1-√2|=?

A. 0
B. 2√2
C. 2√3
D. 9
E. 11
Using the fact that √3 ≈ 1.7 and √2 ≈ 1.4, we have:

|1.7 - 2| + |3 - 1.4| + |5 + 1.7| + |1 - 1.4|

0.3 + 1.6 + 6.7 + 0.4 = 9

Alternate solution:

Recall that |x| = x if x is positive or zero and |x| = -x if x is negative.

Notice that √3-2 < 0, 3 - √2 > 0, 5+√3 > 0 and 1-√2 < 0, we have:

|√3-2|+|3-√2|+|5+√3|+|1-√2|

-(√3 - 2) + 3 - √2 + 5 + √3 - (1 - √2)

-√3 + 2 + 3 - √2 + 5 + √3 - 1 + √2
2 + 3 + 5 - 1

9

Answer: D

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