HeyArnold wrote:Which of the following inequalities is equivalent to the shaded part of the number line shown to the right?

(The number line is x is between -7 and -1 inclusive)

A) |x| > or = -7
B) |x-4| > or = 3
C) |x+4| < or = 3
D) |x| < or = -1
E) |x+3| < or = 4

[spoiler]OA C.... Why is E wrong?! Every point seems to satisfy E as well![/spoiler]

-7 ≤ x ≤ -1 this is what we're looking for


A) |x| ≥ -7

-7 ≤ x ≤ 7


B) |x-4| ≥ 3

x-4 ≥ 3, x ≥ 7


C) |x+4| ≤ 3

x+4 ≤ 3, x ≤ -1

x+4 ≥ -3, x ≥ -7

-7 ≤ x ≤ -1

Some general rules for Absolute Value Inequalities that you should remember:

If the solution is bounded on both sides, like a < x < b, then the answer will look like |?| < number (LESS THAN)

If the solution has two regions, like x < a OR x > b, then the answer will look like |?| > number (GREATER THAN)

Either way, there are two boundary points, a and b. Find their average (aka the number half-way between them). Call this V.
Now, find the distance from either a or b to V. Call this D.

The answer is always |x - V| (LESS THAN OR GREATER THAN) D.

I know this is a lot of words, but applying it doesn't take too long:
Bounded solution means |?| < number
The number half-way between -1 and -7 is -4. The distance from either one to -4 is 3.
|x - V| (LESS THAN OR GREATER THAN) D becomes
|x - -4| < 3 or |x + 4| < 3

Yes, you can plug in, but on the test, chances are good that you are going to have to plug in 3-4 times on this type of question to narrow it down to just one answer. And, as you saw, it's not always obvious WHAT to plug in on this type of question to distinguish between two similar choices.

This is (in my opinion) one case where using the algebra can save you time.

In case of E,
Simply solve for x for both positive and negative of the absolute terms.

|x+3|<=4

for x+3>0,

x+3<=4
=> x<=1 (subtracting 3 from both sides) .......(1)

for x+3<0,
-(x+3)<=4
=> -x-3<=4
=> -x<=7 (Adding 3 on both sides)
=> x>=-7 (Negating both sides and changing sign hence).......(2)

Combining (1) and (2),

-7=<x<=1.

Hence it does not come within -7<=x<=-1 as asked in the question.