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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. anxn, an-1xn-1 +…+a1x + a0 < 0 2. anxn, an-1xn-1 +…+a1x + a0 > 0 3. anxn, an-1xn-1 +…+a1x + a0 < 0 4. anxn, an-1xn-1 +…+a1x + a0 > 0 where an = 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 Notation [a,b] a<x<b (a,b) a<x<b a<x<b [a,b) (a,b] Unbounded intervals are also written in this notation Inequality x > a x>a Notation [a, +∞) (a, +∞) x<b (-∞, b] x<b (-∞, b) Example 1: Solve x3 – 3x2 > 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: (-2, 0) (5, ∞) WORKS Does NOT work WORKS Does NOT work You Try: 1. Solve the inequality algebraically -3x3 + 10x2 < - 8x Example 2: Solve 2x3 + x2 – 6x < 0 by graphing. Graph and find your zeros x = -2, 0, 3/2 (-∞, -2] and [0, 3/2] You Try: 1. Solve the inequality using a graph 2x3 + 8x2 < - 6x