1. 2. 8 Solving Equations in One Variable 2
2.8 Solving Equations in One Variable 2.9 Solving Inequalities in One Variable
After completing this lesson, you should be able to:
Solve equations involving fractions using both algebraic and graphical techniques
Identify extraneous solutions
Solve inequalities involving polynomial and rational functions using both algebraic and graphical techniques.

2. Consider this … What strategy would you use to solve 3 = 2x___?
x x + 1
x___ ≤ 0 ?

3. Rational Equation - equations involving rational expressions or fractions
In most cases we solve rational equations by multiplying by the least common denominator.
If the LCD contains a variable, this may result in an equation of which the solutions are not solutions to the original equation which are referred to as extraneous solutions (really aren’t solutions!!)

4. Ex 1 Solve each equation algebraically
Ex 1 Solve each equation algebraically. Confirm graphically and identify any extraneous solutions.
a) x + 5 = 14 b) x

5. Ex 1 Solve each equation algebraically
Ex 1 Solve each equation algebraically. Confirm graphically and identify any extraneous solutions.
a) = 3 - x x + 2 x2 + 2x x b) 3x + 1 = 7 x + 5 x – 2 x2 + 3x - 10

6. Find the dimensions of the rectangle with minimum perimeter if its area is 200 square meters. Find this least perimeter.

7. Polynomial Inequalities
Every polynomial inequality can be expressed in such a way that f(x) is on one side of an inequality symbol (≤, ≥, >, <)and 0 on the other.
Solving Polynomial Inequalities
To solve f(x) > 0, find values of x which make f(x) positive.
To solve f(x) < 0, find values of x which make f(x) negative.
A sign chart can be an easy way to identify the value of x which solve a polynomial inequality.

8. Ex 1 Determine the x values that cause the polynomial function to be (a) zero (b) positive, and (c) negative
f(x) = (x + 2)(x + 1)2(x – 5)
f(x) = (2x2 + 5)(x – 8)2(x + 1)3

9. Our work in the previous example allows us to report the solutions of four polynomial inequalities:
The solution of f(x) = (x + 2)(x + 1)2(x – 5) > 0 is
The solution of f(x) = (x + 2)(x + 1)2(x – 5) ≥ 0 is
The solution of f(x) = (x + 2)(x + 1)2(x – 5) < 0 is
The solution of f(x) = (x + 2)(x + 1)2(x – 5) ≤ 0 is

10. Important General Characteristics of poly functions and poly inequalities:
Changes sign at its real zeros of odd multiplicity
Touches the x-axis but does not change sign at its real zeros of even multiplicity
Has no x-intercepts or sign changes at its non-real complex zeros associated with irreducible quadratic factors.

11. Graphing using a sign chart
Complete the exploration on p. 259
You have 12 min.
The first 6 min. NO TALKING
You may discuss your results with a neighbor
Prepare to share your results in 2 min.

12. Ex 2 Complete the factoring, if needed, and solve the polynomial inequality using a sign chart. Support graphically.
f(x) = 2x3 – 3x2 – 11x + 6 ≤ 0
f(x) = (x + 1)(x – 3)(x + 2) > 0

13. Ex 3 Solve the polynomial graphically.
2x3 – 5x2 + 3x < 0
x3 – 4x2 – x + 4 ≥ 0

14. Ex 3 Solve the rational inequality using a sign test & support graphically.
x2 + 3x + 2 < 0
x – 1
x3 – 4x2 – x + 4 ≥ 0
2x2 – 3x + 1

15. x3 – 4x2 – x + 4 ≥ 0
2x2 – 3x + 1