## Solving Absolute Value Problems

### Algebra: Inequalities Help

#### Home | Inequalities menu | Algebra menu

First of all, make sure you know how to solve basic inequalities. These types of problems are hard enough as is without being crippled by not knowing the basics. Knowing how to use the distributive law will keep your life easier too.

There are basically 2 types of absolute value problems:

| BLOB | < BLURB

and

| BLOB | > BLURB

We'll call the first one a "lesser-than" problem and the second one a
"greater-than" problem. In both types of problems, there are 2
inequalities you have to solve, giving you 2 solution sets for x. The
bottom line is **the final solution set for x has to contain both
solution sets for x.** Wasn't that a mouthful to comprehend? Don't
worry. Examples are on the way.

In the lesser-than problem, the 2 inequalities you have to solve are:

BLOB < BLURB (when BLOB > 0)

BLOB > -(BLURB) (when BLOB < 0)

In the greater-than problem, the 2 inequalities you have to solve are:

BLOB > BLURB (when BLOB > 0)

BLOB < -(BLURB) (when BLOB < 0)

Enough theory. Let's do some examples.

**Example 1:** Solve:

|x - 3| < 2x + 5

The 2 inequalities we have to solve are:

x - 3 < 2x + 5 (based on x - 3 > 0 or x > 3)

x - 3 > -(2x + 5) (based on x - 3 < 0 or x < 3)

When we solve the first inequality we get x > -8. However, since x >
3, the final solution for the first inequality is

When we solve the second inequality we get x > -2/3. However, since x < 3, the
final solution for the second inequality is

**Example 2:** Solve:

|x + 2| > -3x - 1

The 2 inequalities we have to solve are:

x + 2 > -3x - 1 (based on x + 2 > 0 or x > -2)

x + 2 < -(-3x - 1) (based on x + 2 < 0 or x < -2)

When we solve the first inequality we get x > -3/4. Since x > -2,
the solution for the first inequality is

When we solve the second inequality we get x > 1/2. However, this is
based on the premise that x < -2. Since both conditions can't be true at
the same time, there is no solution for the second inequality.
As before, our final solution set has to contain both. So, our final answer
is

Sometimes, the solution set for x is sandwiched in between 2 numbers.
For example, if one inequality solution is

## Algebra: Inequalities Help

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