1. Solving Equations and Inequalities with Technology

2. y = Left Side and y = Right Side y = 5x – 3 and y = 2 Solve: NOW, consider TWO functions

3. x = 1

5. y = Left Side and y = Right side y = -x + 3 and y = 5x - 3 Solve:

6. y = Left Side and y = Right side y = 5x – 3 and y = 2 Solve:

7. The function corresponding to the L eft S ide is above the function corresponding to the R ight S ide NOTE x < 1 The y-value of the function corresponding to the L eft S ide is greater than the y-value of the function corresponding to the R ight S ide Blue Graph ABOVE Red Graph

8. x = 3 x = -2 Solve:

9. x = 3x = -2 BUT what if...

10. x = 3 x = -2 Now consider... x < -2 ORx > 3 For what values of x is the quadratic ABOVE the linear?

11. x = 3 x = -2 Similarly for... x < -2 OR x > 3

12. Solve:

13. Consider the inequality: x ≤ -1 or x ≥ - 0.8 A “small” gap for -1 < x < - 0.8 …which is the solution to: Red Below Blue x = - 0.8

14. All of the early examples COULD be solved algebraically. Now consider Using technology, the intersection points will be... (-1.06, -0.87) (1.73, 0.99) x = - 1.06 x = 1.73 OR

15. Consider the solution to the corresponding inequality. (1.73, 0.99) (-1.06, -0.87) - 1.06 < x < 1.73 What is the solution for: x 2 > sin(x)?

16. Now consider the solution to: (3.54, 4.21) (-0.30, 0.88) (-2.95, 0.30) x = 3.54 x = -2.95x = -0.30 OR

17. And the inequality: (3.54, 4.21) (-0.30, 0.88) (-2.95, 0.30) x ≥ 3.54 -2.95 ≤ x ≤ - 0.30

18. and there’s much, much more...