This document discusses two methods for solving polynomial inequalities in one variable: 1. Using a sign graph by factoring the polynomial to find zeros, plotting them on a number line, and determining the sign of the polynomial in each interval. 2. Analyzing a graph of the polynomial function using a graphing calculator to find zeros and determine where the graph is above or below the x-axis. Rational inequalities can also be solved using sign graphs, plotting zeros of the numerator and denominator and checking all intervals.

1. 3-2 Polynomial
Inequalities
in One Variable
Objective:
Solve polynomial inequalities in one variable by:
1. Using a sign graph
2. Analyzing a graph of P(x)

2. Polynomial Inequalities
• If P(x) is a polynomial, then P(x) > 0 and
P(x) < 0 are polynomial inequalities.
•We will learn two methods to solve:
1. Using a sign graph
2. Analyzing a graph of P(x)

3. Method 1 – Using a Sign Graph
• Use if the polynomial is factorable.
• Factor to find zeros
• Plot on a number line
• Test values from each interval to
determine the sign of P(x)
• Hint:
• P(x) < 0 means find negative intervals
• P(x) > 0 means find positive intervals

4. Example 1
• Solve x3 – 2x2 – 3x < 0 using a sign graph.

5. You Try!
• Solve: 2x2 + 3x – 5 < 0

6. Example 2
• Solve (x2 – 1)(x – 4)2  0

7. You Try!
• Solve: x4 – 4x2  0

8. Rational Inequalities
• Use same method for rational inequalities
where P(x) and Q(x) are polynomials.
To solve:
• Plot all zeros of numerator and denominator
• Use an open dot for zeros of the
denominator
▫ they make the function undefined: not part of
solution
• Check all intervals, don’t assume signs
alternate!

9. Example 3
• Solve

10. You Try!
• Solve

11. Method 2 – Analyze the Graph
• Useful for functions that are not
factorable.
• Graph the function on the calculator.
• Find zeros using trace
• P(x) > 0 where graph is above x-axis
• P(x) < 0 where graph is below x-axis

12. Example 4
• Solve 2x3 + x2 – 8x + 3 > 0 using a
graphing calculator.

13. You Try!
• Solve 4x3 – 3x2 – 9x – 2  0