Solving Polynomial and Rational Inequalities (and review)

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What is a polynomial inequality?

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A polynomial inequality is an inequality that involves a polynomial expression, typically in the form of P(x) > 0, P(x) < 0, P(x) ≥ 0, or P(x) ≤ 0.

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How do you solve a polynomial inequality?

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1. Rewrite the inequality in standard form. 2. Find the roots of the corresponding polynomial equation. 3. Use test points in the intervals defined by the roots to determine where the inequality holds.

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What is a rational inequality?

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A rational inequality is an inequality that involves a rational expression, typically in the form of \( \frac{P(x)}{Q(x)} > 0 \) or \( \frac{P(x)}{Q(x)} < 0 \), where P(x) and Q(x) are polynomials.

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What is the significance of the roots in solving inequalities?

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The roots of the polynomial or rational expression divide the number line into intervals. The sign of the expression in each interval determines where the inequality is satisfied.

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What is a slant asymptote?

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A slant asymptote is a diagonal line that a graph approaches as x approaches infinity or negative infinity. It occurs when the degree of the numerator is one more than the degree of the denominator.

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How do you find the slant asymptote of a rational function?

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To find the slant asymptote of a rational function \( \frac{P(x)}{Q(x)} \), perform polynomial long division. The quotient (ignoring the remainder) gives the equation of the slant asymptote.

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What does it mean for a function to be continuous at a point?

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A function is continuous at a point x = c if: 1. f(c) is defined, 2. \( \lim_{x \to c} f(x) \) exists, and 3. \( \lim_{x \to c} f(x) = f(c) \).

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What are the types of discontinuities?

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1. Infinite Discontinuity: The function approaches infinity at a point. 2. Removable Discontinuity: The function is not defined at a point but can be made continuous. 3. Jump Discontinuity: The function has a sudden change in value at a point.

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What is the difference between strict and non-strict inequalities?

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Strict inequalities (>, <) do not include the boundary points, while non-strict inequalities (≥, ≤) include the boundary points.

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How do you express the solution set of an inequality?

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The solution set of an inequality can be expressed in interval notation, set-builder notation, or graphically on a number line.