Quadratic inequalities can be written in the form ax 2 + bx + c
< 0, ax 2 + bx + c 0, ax 2 + bx + c > 0, or ax 2 + bx + c 0.
The solutions to quadratic inequalities are written as intervals.
An interval is the set of all real numbers between two given
numbers. The two numbers on the ends are the endpoints. The
endpoints might or might not be included in the interval depending
on whether the interval is open, closed, or half- open. 5.2.5:
Solving Quadratic Inequalities2
Slide 4
The solutions to a quadratic inequality can be one interval or
two intervals. Use these solutions to create regions on a number
line and test points in each region to solve the inequality. If the
quadratic equation has only complex solutions, the expression is
either always positive or always negative. In these cases, the
inequality will have no solution or infinitely many solutions.
5.2.5: Solving Quadratic Inequalities3
Slide 5
Solutions of quadratic inequalities are often graphed on number
lines. The endpoints of the solution interval are represented by
either an open dot or a closed dot. Graph the endpoints as an open
dot if the original inequality symbol is. Graph endpoints as a
closed dot if the original inequality symbol is or . 5.2.5: Solving
Quadratic Inequalities4
Slide 6
For what values of x is ( x 2)( x + 10) > 0? 5.2.5: Solving
Quadratic Inequalities5
Slide 7
The expression will be positive when both factors are positive
or both factors are negative. 5.2.5: Solving Quadratic
Inequalities6
Slide 8
x 2 is positive when x > 2. x + 10 is positive when x >
10. Both factors are positive when x > 2 and x > 10, or when
x > 2. 5.2.5: Solving Quadratic Inequalities7
Slide 9
x 2 is negative when x < 2. x + 10 is negative when x <
10. Both factors are negative when x < 2 and x < 10, or when
x < 10. ( x 2)( x + 10) > 0 when x > 2 or x < 10.
5.2.5: Solving Quadratic Inequalities8
Slide 10
Solve x 2 + 8 x + 7 0. Graph the solutions on a number line.
5.2.5: Solving Quadratic Inequalities9