1140 L18 - Lecture 18: Section 2.7 Nonlinear Inequalities...

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Lecture 18: Section 2.7 Nonlinear Inequalities To Solve a Polynomial Inequality 1. Write the inequality with 0 on the right side to obtain one of the following: f ( x ) > 0 f ( x ) ± 0 f ( x ) < 0 f ( x ) ² 0 2. Find all real zeros of the polynomial f ( x ). These zeros are the critical numbers of f ( x ). 3. Use the critical numbers to separate the number line into test intervals . 4. Choose a number a in each interval and evaluate f ( a ): a) If f ( a ) > 0, then f ( x ) > 0 in that interval. b) If f ( a ) < 0, then f ( x ) < 0 in that interval. 5. Include endpoints if f ( x ) ± 0 or if f ( x ) ² 0. The solution set of the inequality is the set of all ordered pairs ( x; y ) that satisfy the inequality.
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FACT: A polynomial changes signs only at its zeros. Between two zeros, a polynomial must be entirely positive or entirely negative. ex. Solve the inequalities: 2 x 2 ± 3 x > 5. Write your answer in interval notation.
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It is easier to use the factored form of the polynomial to determine the signs of the polynomial in each in- terval.
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1140 L18 - Lecture 18: Section 2.7 Nonlinear Inequalities...

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