Problem Solving

Brent@GMATPrepNow wrote:

greenwich wrote: Brent, for the 3 steps to solve the 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 in the solutions to check for extraneous roots

Does #3 Plug in the solutions to check for extraneous roots also applies to Absolute Value with Inequality such as II in the above question (|x+3|>2)?

Hi greenwich,

I believe you're referring to the advice I gave in this post: https://www.beatthegmat.com/is-x-0-t273441.html

That advice deals with equations with absolute value, not inequalities.

Cheers,
Brent

Brent@GMATPrepNow wrote:

greenwich wrote:If 4<[(7-x)/3], which of the following must be true?
I. 5<x
II. |x+3|>2
III. -(x+5) is positive

A) II only
B) III only
C) I and II only
D) II and III only
E) I, II and III

First, let's deal with the given inequality.
4 < (7-x)/3
Multiply both sides by 3 to get: 12 < 7 - x
Add x to both sides: x + 12 < 7
Subtract 12 from both sides to get: x < -5

So, if x < -5, which of the following statements MUST be true?

Aside: When dealing with "MUST be true" questions, we can eliminate a statement if we can find an instance where it is not true.

I. 5 < x (MUST this be true?)
No!
If x < -5, then it could be the case that x = -7, and -7 is NOT greater than 5
So, statement I need NOT be true.

II. |x+3| > 2 (MUST this be true?)
The answer is Yes. Here's why:
IMPORTANT CONCEPT: |x - k| represents the DISTANCE between x and k on the number line.
So, for example, we can think of |4 - 7| as the distance between 4 and 7 on the number line.
Notice that |4 - 7| = |-3| = 3, and 3 is indeed the distance between 4 and 7 on the number line.

Now let's examine |x+3|
We can rewrite this as |x - (-3)|
This represents the DISTANCE between x and -3 on the number line.
So, the inequality |x-(-3)| > 2 is stating that the DISTANCE between x and -3 on the number line is GREATER THAN 2
Well, since we're told that x < -5, we can be certain that the DISTANCE between x and -3 on the number line is definitely GREATER THAN 2
[If you're not convinced, sketch a number line, and place a big dot at -3. Then choose ANY value for x such that x < -5. You'll see that the distance between x and -3 is greater than 2]
So, statement II MUST be true.

III. -(x+5) is positive
This is the same as saying -(x+5) > 0 (MUST this be true?)
The answer is Yes. Here's why:
We're told that x < -5
If we add 5 to both sides we get x+5 < 0
Now, if we multiply both sides by -1, we get -(x+5) > 0
[aside: notice that, since I multiplied both sides by a negative value, I reversed the direction of the inequality]
As we can see, statement III MUST be true.

Answer: D

Cheers,
Brent

If 4 < (7-x)/3,which of the following must be true?

• I. 5<x
  2. |x+3|>2
  3. -(x+5) is positive.
  
  A. II only
  B .III only
  C. I and II only
  D. II and III only
  E. I,II and III

Brent@GMATPrepNow wrote: