Nonlinear Inequalities MATH 109 - Precalculus S. Rook

Overview • Section 2.7 in the textbook: – Solving polynomial inequalities 2

Solving Polynomial Inequalities

Solving Inequalities • • • An inequality is similar to an

equality

except instead of =, we have >, <, ≤ or ≥.

– Essentially solved in the same way as an

equality Solution set

– all values that satisfy an inequality Whereas an

equality

has at most 1 solution, the solution to an inequality is a set – possibly with infinitely many elements 4

Solving Polynomial Inequalities • • • We discussed how to solve polynomial

equalities

Only difference is that the

solution set

consists of

intervals

number now instead of a single real To solve an inequality such as (x + 1)(x – 1) > 0: – By the

Zero Product Principle

, x = -1 or x = 1 – This subdivides the interval (-oo, +oo) into three subintervals: (-oo, -1), (-1, 1), (1, +oo) 5

Solving Polynomial Inequalities (Continued) – Pick one value in each subinterval and test it in the inequality to find the

sign

of that subinterval • Recall the

sign property of polynomials

– The sign in each subinterval is the

SAME

• Keep only those intervals that satisfy the inequality • If more than one interval satisfies the inequality, union them 6

Solving Polynomial Inequalities (Example)

Ex 1:

Write the solution set in interval notation: a) x 2 + 4x + 4 ≥ 9 c) 2x 2 – 7x ≤ -2 b) 3x 2 + 7x < 6 d)

x

3 – 4x > 0 7

Summary • • • After studying these slides, you should be able to: – Solve polynomial inequalities, graph the solution set on a number, and write the solution set in interval notation Additional Practice – See the list of suggested problems for 2.7

Next lesson – Exponential Functions and Their Graphs (Section 3.1) 8