Quadratic Inequalities Examples

Example 1
Solve the inequality 3x ^{2} – 8x + 4 > 0.

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

(3x – 2)(x – 2) > 0

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The roots are and x = 2.

Putting the roots on the number line and evaluating each regions shows us that the equation is positive when and 2 < x < ∞.

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Example 2
Solve the inequality -x ^{2} – 4x > 3.

If things have to be unequal, they may as well be unequal for everyone.

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

So start off by putting everything on the same side of the inequality.

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 equation is negative from -4.3 < x < -0.7. Maybe if it had a lollypop it would feel better?

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Example 3
Solve the inequality -2x ^{2} + 8x + 8 ≤ 0.

Be careful here.

x ^{2} – 4x – 4 ≥ 0

You want to get rid of the -2 from the equation, but you have to reverse the inequality when you do.

Now we search for the roots.

and

These are about x = -0.8 and x = 4.8.

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

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