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If |x - 2| = |x + 3|, x could equal

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If |x - 2| = |x + 3|, x could equal

by M7MBA » Thu Jun 14, 2018 11:08 pm

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If |x - 2| = |x + 3|, x could equal

A. -5
B. -1/2
C. 1/2
D. 5
E. No real solution

The OA is the option B.

Could someone tell me what is the best way of solving this PS question? Should I try number by number? I'd be thankful for your help.
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by Vincen » Fri Jun 15, 2018 12:16 am
Hello M7MBA.

Well, you can plug in the values and see which option is the correct.

Also, we can solve it by expanding the absolute values |x - 2| and |x + 3| as follows:

Each expression between the absolute values is equal to zero when x=2 and x=-3. Then, we have the following cases:

Case 1: x<-3.

- If x<-3 then x+3<0.
- If x<-3 then x<2 and therefore x-2<0.

Hence $$\left|x-2\right|=\left|x+3\right|\ \ \Leftrightarrow\ \ \ -\left(x-2\right)=-\left(x+3\right)\ \ \ \Leftrightarrow\ \ -x+2=-x-3\ \ \ \Leftrightarrow\ \ 2=-3\ \ \ False.$$ In this case we didn't get a real solution.

Case 2: -3 < x < 2.

- If -3<x then x+3>0.
- If x < 2then x-2<0.

Hence $$\left|x-2\right|=\left|x+3\right|\ \ \Leftrightarrow\ \ \ -\left(x-2\right)=x+3\ \ \ \Leftrightarrow\ \ -x+2=x+3\ \ \ \Leftrightarrow\ \ 2x=-1\ \ \ \Leftrightarrow\ \ \ x=-\frac{1}{2}.$$ This implies that the correct answer is the option [spoiler]C=-1/2[/spoiler].

Finally

Case 3: x > 2

- If x > 2 then x-2 > 0.
- If x > 2 then x > -3 and therefore x+3 > 0.

Hence $$\left|x-2\right|=\left|x+3\right|\ \ \Leftrightarrow\ \ \ x-2=x+3\ \ \ \Leftrightarrow\ \ -2=3\ \ False$$ I hope it helps you.
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by GMATGuruNY » Fri Jun 15, 2018 2:05 am
M7MBA wrote:If |x - 2| = |x + 3|, x could equal

A. -5
B. -1/2
C. 1/2
D. 5
E. No real solution
Case 1: signs unchanged
x-2 = x+3
-2 = 3
Since the resulting equation is invalid, Case 1 is not possible.

Case 2: signs changed in ONE of the absolute values
x-2 = -x-3
2x = -1
x = -1/2.

The correct answer is B.
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by GMATGuruNY » Fri Jun 15, 2018 2:09 am
M7MBA wrote:If |x - 2| = |x + 3|, x could equal

A. -5
B. -1/2
C. 1/2
D. 5
E. No real solution
Alternate approach:
|a-b| = the distance between a and b.
|a+b| = |a-(-b)| = the distance between a and -b.

|x - 2| = |x + 3| implies the following:
The distance between x and 2 is equal to the distance between x and -3.
In other words, x must be HALFWAY BETWEEN -3 AND 2.
Halfway between two numbers = the AVERAGE of the two numbers:
(-3+2)/2 = -1/2.

the correct answer is B.
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by Brent@GMATPrepNow » Fri Jun 15, 2018 5:13 am
M7MBA wrote:If |x - 2| = |x + 3|, x could equal

A. -5
B. -1/2
C. 1/2
D. 5
E. No real solution
There are 3 steps to solving equations involving ABSOLUTE VALUE:
1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug solutions into original equation to check for extraneous roots

Given: |x - 2| = |x + 3|
case a: x - 2 = x + 3
Subtract x from both sides to get: -2 = 3
No good.
So, this equation has no solution.

case b: x - 2 = -(x + 3)
Simplify: x - 2 = -x - 3
Add x to both sides: 2x - 2 = -3
Add 2 to both sides: 2x = -1
Solve: x = -1/2

IMPORTANT: Plug x = -1/2 into the ORIGINAL equation, to make sure it isn't an extraneous root.
We get: |-1/2 - 2| = |-1/2 + 3|
Evaluate: |-2.5| = |2.5|
Evaluate: 2.5 = 2.5
Works!
So, x = -1/2 IS a solution.

Answer: B

Cheers,
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by GMATGuruNY » Fri Jun 15, 2018 5:24 am
IMPORTANT: Plug x = -1/2 into the ORIGINAL equation, to make sure it isn't an extraneous root.
The warning above must be heeded when only one side of an equation is an absolute value.
But when BOTH sides of an equation are absolute values, any solution that we derive will be guaranteed to work, so the step above is unnecessary.
Since both sides of |x - 2| = |x + 3| are absolute values, we do not need to check the validity of x=-1/2.
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by Jeff@TargetTestPrep » Sun Jun 24, 2018 5:02 pm
M7MBA wrote:If |x - 2| = |x + 3|, x could equal

A. -5
B. -1/2
C. 1/2
D. 5
E. No real solution
By scanning the answer choices, we see that when x = -1/2, we have

|-1/2 - 2| = |-5/2| = 5/2 and |-1/2 + 3| = |5/2| = 5/2

Alternate Solution:

We consider two cases for this absolute value question.

Case 1. Drop the absolute value symbols and solve for x.

x - 2 = x + 3

0 = 5

This doesn't work.

Case 2. (x - 2) is positive and (x + 3) is negative.

x - 2 = -(x + 3)

x - 2 = -x - 3

2x = -1

x = -½

When x = -1/2, we see that |x - 2| = |x + 3| becomes |(-1/2) - 2| = |(-1/2) + 3|, or 5/2 = 5/2, which is a true statement. Thus, the solution for this equation is x = -1/2.

Answer: B

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