Help with absolute-value equations and inequalities?

Question

I've been trying to work on these problems all day, and I just can't figure them out! Here they are. Thank you.

7 | q | - 2 = 9

5 - 2 | 3x - 4 | = -5

| x | < 2 I think this one is { x | -2 < x < 2 } ?

| x - 1| < 4

| x + 2 | is smaller than or equal to 6

| x - 3 | + 2 > 7

| 2y - 9 | > -5

Answer 1

The key to handling absolute values, is to solve to get the expression inside the || alone, then solve for each possible value of the expression.

1.) 7 | q | - 2 = 9
Solve like a normal equation:
7 | q | - 2 = 9
7 | q | = 11
| q | = 11/7
q = +/- 11/7

2.) 5 - 2 | 3x - 4 | = -5
Solve to get 3x - 4 alone:
- 2 | 3x - 4 | = -10
2 | 3x - 4 | = 10
| 3x - 4 | = 5

Now, solve for each possibility of | 3x - 4 |.
3x - 4 = 5
3x = 9
x = 3
3x - 4 = -5
3x = -1
x = -1/3

So we write x = 3, -1/3.

3.) | x | < 2 I think this one is { x | -2 < x < 2 }
You're correct.

4.) | x + 2 | <= 6
Split it up and solve for both values of |x + 2|:
x + 2 <= 6
x <= 4

x + 2 >= -6 (sign switches when you go negative)
x >= -8

So we write: -8 <= x <= 4