Solve each rational inequality. Give the solution set in interval notation. (2x - 3)/(x + 1) > 4

검증된 단계별 안내

Start by rewriting the inequality to have zero on one side: \( \frac{2x - 3}{x + 1} - 4 > 0 \).

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

Simplify the numerator: \( 2x - 3 - 4x - 4 = -2x - 7 \), so the inequality becomes \( \frac{-2x - 7}{x + 1} > 0 \).

Determine the critical points by setting the numerator and denominator equal to zero: \( -2x - 7 = 0 \) and \( x + 1 = 0 \). Solve these to find \( x = -\frac{7}{2} \) and \( x = -1 \).

Use these critical points to divide the number line into intervals, then test a value from each interval in the inequality \( \frac{-2x - 7}{x + 1} > 0 \) to determine where the inequality holds true. Express the solution set in interval notation, remembering to exclude points where the denominator is zero.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

도움이 되었나요?

주요 개념

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

Rational Inequalities

Rational inequalities involve expressions where one rational expression is compared to another using inequality symbols. Solving them requires finding values of the variable that make the inequality true, often by analyzing the sign of the numerator and denominator separately.

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 where the rational expression is positive or negative, which helps identify the solution set.

Interval Notation

Interval notation is a concise way to express solution sets of inequalities using parentheses and brackets to indicate open or closed intervals. It clearly shows the range of values that satisfy the inequality, excluding points where the expression is undefined.