a quadratic inequality is an inequality in the form ax 2 +bx+c >,

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Page 1: A quadratic inequality is an inequality in the form ax 2 +bx+c >,
Page 2: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

A quadratic inequality is an inequality in the form ax2+bx+c >, <, ≤, or ≥ 0.

In most cases, you just need to find the points where ax2+bx+c = 0 and use these points as the boundaries of your solution set.

Page 3: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

Unfortunately, you won’t always have two solutions to the equation ax2+bx+c = 0.

If you have only one solution, there are several possibilities.

If you have no solutions, the solution is either the empty set or all x.

Page 4: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

Let’s look at the case where ax2+bx+c = 0 at exactly one point.

There are several possibilities for solutions to these inequalities, depending on the type of inequality.

These situations are summarized on a table on the next page.

Page 5: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

This table assumes that a > 0. If a < 0, multiply both sides of your equation by -1 and flip the inequality.

Rather than memorizing this, try to understand why each one is the way it is.

Type of Inequality Type of Solution

ax2+bx+c > 0 Single exception

ax2+bx+c < 0 No solution

ax2+bx+c ≥ 0 All real numbers

ax2+bx+c ≤ 0 Single solution

Page 6: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

Say we’re trying to solve the inequalityx2 – 2x + 1 ≤ 0.

We start by finding the points where x2 – 2x + 1 = 0, as usual. Factoring the equation, we have (x -1)2 = 0.

This tells us that we have only one point where x2 – 2x + 1 = 0: x = 1.

Since the x2 term is positive, x will be greater than 0 at all other points.

This means that we have only one solution to our inequality: x = 1.

Page 7: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

Note that x2 – 2x + 1 = 0 at exactly one point and x2 – 2x + 1 > 0 at all other points. If we had instead been solving x2 – 2x + 1 < 0, our solution would have been the empty set. In general, graphing the equation makes it easier to tell what your solution set is. Just look for where the equation fulfills the inequality.

Page 8: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

If x is never equal to zero, there are two possibilities.

Either the inequality is true for all x, or the inequality is false for all x.

To figure out which type of situation you have, all you have to do is look at the type of inequality you have.

Page 9: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

First, make sure a > 0. If a < 0, multiply both sides of your equation by -1 and flip the inequality.

Afterwards, if you have an equality where ax2+bx+c < 0 or ax2+bx+c ≤ 0, the solution to the inequality is the empty set. If you have an inequality where ax2+bx+c > 0 or ax2+bx+c ≥ 0, the solution to the inequality is all real numbers.

Page 10: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

Say we’re trying to solve the inequality -x2 - 5 < 0.

Multiplying by -1, we can simplify this inequality to x2 + 5 > 0.

Notice that x2 +5 ≠ 0 for any real x. This means that we can determine our

solution set based on the type of inequality we’re working with.

We have a “greater than zero” inequality, so our solution set is all real numbers.

Page 11: A quadratic inequality is an inequality in the form ax 2 +bx+c >,

Here is a graph of x2 + 5. Note that it’s greater than 0 for all x.