Problem Solving

tonebeeze wrote:Quant Rev #156: Inequalities

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

Simplifying the given inequality we have:

4 < (7 - x)/3

12 < 7 - x

5 < -x

-5 > x

Since x is less than -5, we see that |x+3| > 2.

Also, since x < -5, we see that x + 5 will always be negative and thus -(x+5) will always be positive.

Answer: D

tonebeeze wrote:Quant Rev #156: Inequalities

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

OA: D

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