1. 5.7 : Graphing and Solving Quadratic Inequalities
Objectives: Students will be able to…
Graph quadratic inequalities in 2 variables
Solve quadratic inequalities in 1 variable

2. Quadratic Inequalities in 2 variables…
The graph contains all solutions (x,y) that make the inequality true

3. Graphing a Quadratic Inequality in 2 variables:
Graph the parabola with the equation y=ax2 + bx +c (dashed for > or < ; solid line for > or <)
Choose a point (x,y) inside the parabola and check whether the point is a solution of the inequality
If the point is a solution, shade the region inside the parabola. If it is not a solution, shade the region outside the parabola.

4. Example
Graph y < 2x2 – 5x -3

5. Graph y > - x2 +5

6. Graph y>x2 – 4x +3

7. Systems of Quadratic Inequalities
Graph each inequality one at a time on the same coordinate plane
Where the shading overlaps is your solution region. Darken this area.

8. Example:
y < - x2 + 9 y > x2 +5x -6

9. Example:
y < - 3x2 y > (-1/2)x2 - 5

10. Quadratic Inequalities in 1 variable
ax2 + bx + c > 0 ( or >)
ax2 + bx + c < 0 ( or <)
Two ways to solve:
Graphically
Algebraically

11. How to solve graphically
Graph y = ax2 + bx +c (make sure to set=0 first)
Be sure to identify the x – intercepts (may need to use Quadratic Formula…uh oh!!)
If ax2 + bx + c < 0 ( or <), you are looking for the x values where the graph is below the x axis. ( or on or below)
If ax2 + bx + c > 0 ( or >), you are looking for the x values where the graph is above the x axis. ( or on or above)

12. Solve by graphing: x2 – 5x + 6 > 0
Where is your graph above the x-axis?

13. Solve by graphing: x2 – 11x + 5 < 0
Where is your graph below the x – axis?

14. Solving 1 variable inequalities algebraically
Solve 2x2 – x > 3
Write the corresponding equation:
2x2 – x = 3
2. Solve the equation:
3. Test these critical x values on a number line. Pick x values to the left and right of the critical values to see what numbers satisfy the inequality:

15. Solve algebraically.
1. 2x2 < 8
2. x2 – 10x + 24 > 0

16. Ticket out: Solve 16 – x2 > 0

17. Remember this material when you are finding domain and range in precalculus and calculus!!!!