2.4 Solve Polynomial Inequalities Inequalities Test: 2/26/10

Vocabulary A polynomial inequality in one variable can be written as one of the following: 1. a n x n, a n-1 x n-1 +…+a 1 x + a 0 < 0 2. a n x n, a n-1 x n-1 +…+a 1 x + a 0 > 0 3. a n x n, a n-1 x n-1 +…+a 1 x + a 0 < 0 4. a n x n, a n-1 x n-1 +…+a 1 x + a 0 > 0 where a n = 0

Vocabulary Inequalities can be used to describe subsets of real numbers called intervals.  In bounded intervals below, the real numbers a and b are endpoints of each interval. Inequality a<x<b a<x<b a<x<b a<x<b Notation [a,b] (a,b) [a,b) (a,b]  Unbounded intervals are also written in this notation Inequality x > a x > a x < b x < b Notation [a, +∞) (a, +∞) (-∞, b] (-∞, b)

Example 1: Solve x 3 – 3x 2 > 10x algebraically  Step 1: Rewrite as an equation, and put in standard form.  Step 2: Solve for x (we can factor)

Example 1 Cont’d: Plotting our found x-values will help us find our intervals. This breaks our number line into 4 intervals. Test an x-value in each interval to see if it satisfies the inequality

Example 1 Cont’d: Does NOT workWORKS Does NOT work (-2, 0) (5, ∞ )

You Try: Solve the inequality algebraically 1. -3x x 2 < - 8x

Example 2: Solve 2x 3 + x 2 – 6x < 0 by graphing.  Graph and find your zeros  x = -2, 0, 3/2 (-∞, -2] and [0, 3/2]

You Try: Solve the inequality using a graph 1. 2x 3 + 8x 2 < - 6x