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.

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

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!!)

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

Ex 1 Solve each equation algebraically Ex 1 Solve each equation algebraically. Confirm graphically and identify any extraneous solutions. a) 3 + 6 = 3 - x x + 2 x2 + 2x x b) 3x + 1 = 7 x + 5 x – 2 x2 + 3x - 10

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

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.

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

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

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.

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.

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

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

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

x3 – 4x2 – x + 4 ≥ 0 2x2 – 3x + 1