Solving Equations and Inequalities with Technology

y = Left Side and y = Right Side

y = 5x – 3 and y = 2

Solve: 235 x

Q.E.D. R.S.

2

2 R.S. 3 - 5(1) ..

:

1

55

SL

Checking

x

xNOW, consider TWO functions

x = 1

235 yandxy

235 x

353 xx

x = 1

353 xx

The function corresponding

to the Left Side

is above the function corresponding

to the Right Side

NOTE

x < 1

The y-value of the function corresponding to the

Left Side

is greater than the y-value of the function

corresponding to the

Right Side

Blue Graph ABOVE

Red Graph

172 xx

x = 3

x = -2

Solve:

x = 3

x = -2

1x7x2 Now consider . . .

x < -2 OR x > 3

For what values of x is the quadratic is ABOVE the linear?

Solve:

5

4x

51

1

x

Consider the inequality:

51

1

x

x ≤ -1 or x ≥ - 0.8

A “small” gap for -1 < x < - 0.8

…which is the solution to:

51x

1

Red Below Blue

x = - 0.8

All of the early examples COULD be solved algebraically. Now consider

)xsin(x 2

Using technology, the intersection points will be . . .

(-1.06, -0.87)

(1.73, 0.99)

x = - 1.06

x = 1.73OR

Consider the solution to the corresponding inequality.

)xsin(x 2

(1.73, 0.99)

(-1.06, -0.87)

- 1.06 < x < 1.73

What is the solution for: x2 > sin(x)?

Now consider the solution to:x

x

2

32 3

x

x

2

32 3

x

x

2

32 3

x

3

2

3x9x

3

1

(3.54, 4.21)

(-0.30, 0.88)(-2.95, 0.30)

x = 3.54x = -2.95 x = -0.30OR OR

And the inequality:x

x

2

32 3

x

x

2

32 3

x

x

2

32 3

x

xx

2

39

3

1 3

(3.54, 4.21)

(-0.30, 0.88)

(-2.95, 0.30)

x ≥ 3.54-2.95 ≤ x ≤ - 0.30

and there’s much, much more . . .