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. (3x+5)/(6−2x)≥0

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Identify the rational inequality: \(\frac{3x+5}{6-2x} \geq 0\).

Find the critical points by setting the numerator and denominator equal to zero separately: solve \(3x + 5 = 0\) and \(6 - 2x = 0\).

Determine the values of \(x\) where the expression is undefined (denominator zero) and where the expression equals zero (numerator zero). These points divide the number line into intervals.

Test a value from each interval in the original inequality to determine if the expression is positive or negative in that interval.

Based on the test results, write the solution set including points where the expression equals zero (if inequality is \(\geq\)) and exclude points where the expression is undefined, then express the solution in interval notation.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the expression is positive, negative, or zero, considering the domain restrictions where the denominator is not zero.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator 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.

Interval Notation and Graphing Solutions

Interval notation concisely represents sets of real numbers satisfying the inequality, using parentheses for excluded endpoints and brackets for included ones. Graphing on a number line visually shows these intervals, aiding in understanding the solution set.