Solving Equations and Inequalities with Technology

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

x = 1

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

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

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

x = 3 x = -2 Solve:

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

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

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

Solve:

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

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 = x = 1.73 OR

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

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

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

and there’s much, much more...