Solve each rational inequality. Give the solution set in interval notation. (x+3)/(2x-5)≤1

검증된 단계별 안내

Start by rewriting the inequality \( \frac{x+3}{2x-5} \leq 1 \) so that one side is zero. Subtract 1 from both sides to get \( \frac{x+3}{2x-5} - 1 \leq 0 \).

Combine the terms on the left side over a common denominator: \( \frac{x+3}{2x-5} - \frac{2x-5}{2x-5} \leq 0 \). This simplifies to \( \frac{x+3 - (2x-5)}{2x-5} \leq 0 \).

Simplify the numerator: \( x + 3 - 2x + 5 = -x + 8 \). So the inequality becomes \( \frac{-x + 8}{2x - 5} \leq 0 \).

Identify the critical points by setting numerator and denominator equal to zero: numerator zero at \( -x + 8 = 0 \) which gives \( x = 8 \), and denominator zero at \( 2x - 5 = 0 \) which gives \( x = \frac{5}{2} \). These points divide the number line into intervals.

Test values from each interval in the inequality \( \frac{-x + 8}{2x - 5} \leq 0 \) to determine where the expression is less than or equal to zero. Remember to exclude values where the denominator is zero, and include points where the numerator is zero if the inequality allows equality.

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 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 expression is defined and the sign of the numerator and denominator.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, dividing the number line into intervals. By testing values in each interval, you determine where the rational expression is positive, negative, or zero, which helps identify the solution set for the inequality.

Interval Notation

Interval notation is a concise way to express sets of numbers that satisfy inequalities. It uses parentheses for values not included (open intervals) and brackets for values included (closed intervals), clearly showing the range of solutions on the number line.