Solving Polynomial and Rational Inequalities: Zeros, Test Intervals, and Procedure - Prof., Study notes of Trigonometry

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Equations and Inequalities

1.6 Polynomial And Rational Inequalities

August 28, 2009

Definition

Zeros of a polynomial are the values of x that make the polynomial

equal to zero. These zeros divide the real number line into test intervals

where the value of the polynomial is either positive or negative.

For example, consider the polymial x

+ x − 2. Its zeros are

x

+ x − 2 = 0

(x + 2)(x − 1) = 0

x = − 2 or x = 1

Thus the zeros are x = −2 and x = 1. These zeros divide the real

number line into three test intervals:

Procedure for Solving Polynomial Inequalities

I

Step 1: Write the inequality in standard form.

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Step 2: Identify zeros.

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Step 3: Draw the number line with zeros labeled.

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Step 4: Determine the sign of the polynomial in each interval.

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Step 5: Identify which interval(s) make the inequality true.

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Step 6: Write the solution in interval notation.

Example (1)

Solve the inequality x

− x > 12.

The solution is (−∞, −3)(4, ∞).

Example (2)

Solve the inequality x

The solution is [− 2 , 2].

Example (3)

Solve the inequality x

+ 2x ≥ −3.

The solution is (−∞, ∞).