The first step of solving an inequality is to find the roots.

(3x – 2)(x – 2) > 0

Here's a haiku to get you in the mood:

Like the sun's bright rays Finding this factorable Feels super awesome

The roots are and x = 2.

Putting the roots on the number line and evaluating each region shows us that the expression is positive when:

and 2 < x < ∞

Example 2

Solve the inequality -x^{2} – 5x > 3.

If things have to be unequal, they may as well be unequal for everyone. So start off by putting everything on the same side of the inequality. Don't forget to reverse the direction of that inequality sign when we divide everything by -1.

x^{2} + 5x + 3 < 0

The next step is to put on some gloves, find a tree, bush, shrub, or
other plant, and look for roots. The quadratic formula can be a big help
in this.

and

These are about x = -0.7 and x = -4.3.

The number line shows that the expression x^{2} + 5x + 3 is negative from -4.3 < x < -0.7. Maybe if it had a lollipop it would feel better?

Example 3

Solve the inequality -2x^{2} + 8x + 8 ≤ 0.

Be careful here. You want to divide everything by -2, but you need to reverse the inequality when you do.

x^{2} – 4x – 4 ≥ 0

Now we search for the roots.

and

These are about x = -0.8 and x = 4.8. We'll plug our x-values into the original inequality -2x^{2} + 8x + 8 ≤ 0, which means we want the negative values.

So our negative values are from -∞ < x < -0.8 and 4.8 < x < ∞