Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problema 57

Capitolo 4, Problema 57

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x + 4)/(2x - 1) ≤ 3

Guida verificata passo dopo passo

Start by rewriting the inequality to have zero on one side: subtract 3 from both sides to get \(\frac{\!x + 4}{2x - 1} - 3 \leq 0\).

Find a common denominator and combine the terms into a single rational expression: \(\frac{\!x + 4}{2x - 1} - \frac{3(2x - 1)}{2x - 1} \leq 0\), which simplifies to \(\frac{\!x + 4 - 3(2x - 1)}{2x - 1} \leq 0\).

Simplify the numerator by distributing and combining like terms: \(x + 4 - 6x + 3 = -5x + 7\), so the inequality becomes \(\frac{-5x + 7}{2x - 1} \leq 0\).

Identify critical points by setting numerator and denominator equal to zero separately: solve \(-5x + 7 = 0\) and \(2x - 1 = 0\) to find values where the expression changes sign or is undefined.

Use the critical points to divide the number line into intervals, then test a value from each interval in the inequality \(\frac{-5x + 7}{2x - 1} \leq 0\) to determine where the inequality holds true. Finally, express the solution set in interval notation and graph it on the real number line.

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Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to a number or another expression. Solving them requires finding values of the variable that make the inequality true, considering where the denominator is zero to avoid undefined expressions.

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Nonlinear Inequalities

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. By testing points in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

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Point-Slope Form

Interval Notation and Graphing Solutions

Interval notation expresses the solution set compactly using parentheses and brackets to indicate open or closed intervals. Graphing on a number line visually represents these intervals, showing where the inequality is satisfied and highlighting excluded points like zeros of the denominator.

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Interval Notation

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