4.9 – Graph and Solve Quadratic Inequalities

A quadratic inequality in two variables can be written in one of the following forms:

y < ax2 + bx + c

y < ax2 + bx + c

y > ax2 + bx + c

y > ax2 + bx + c

The graph of any such inequality consists of all solutions (x, y) of the inequality.

4.9 – Graph and Solve Quadratic Inequalities

4.9 – Graph and Solve Quadratic Inequalities

Example 1a: Graph a quadratic inequality

Graph y > x2 + 3x – 4

4.9 – Graph and Solve Quadratic Inequalities

Example 1b: Graph a quadratic inequality

Graph y < -x2 – 4x

4.9 – Graph and Solve Quadratic Inequalities

Example 2:

A manila rope used for rappelling down a cliff can safely support a weight W (in pounds)

provided W < 1480d2

where d is the rope’s diameter (in inches). Graph the inequality.

4.9 – Graph and Solve Quadratic Inequalities

Example 3:

Graph the system of quadratic inequalities.

y < -x2 + 4

y > x2 – 2x – 3

4.9 – Graph and Solve Quadratic Inequalities

Example 3:

Graph the system of quadratic inequalities.

y < -x2 + 4

y > x2 – 2x – 3

4.9 – Graph and Solve Quadratic Inequalities

A quadratic inequality in one variable can be written in one of the following forms:

ax2 + bx + c < 0

ax2 + bx + c < 0

ax2 + bx + c > 0

ax2 + bx + c >0

You can solve quadratic inequalities using tables, graphs, or algebraic methods.

4.9 – Graph and Solve Quadratic Inequalities

Example 4:

Solve x2 + x < 6 using a table.

4.9 – Graph and Solve Quadratic Inequalities

Example 4b:

Solve x2 + 2x > 8 using a table.

4.9 – Graph and Solve Quadratic Inequalities

Example 5:

Solve 2x2 + x – 4 > 0 by graphing.

4.9 – Graph and Solve Quadratic Inequalities

Example 5b:

Solve ½x2 + 3x – 2 < 0 by graphing.

4.9 – Graph and Solve Quadratic Inequalities

Example 6:

Solve x2 – 2x > 15 by algebraically.