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

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Start by rewriting the inequality \( \frac{3}{x-2} < 1 \) to have zero on one side. Subtract 1 from both sides to get \( \frac{3}{x-2} - 1 < 0 \).

Find a common denominator to combine the terms on the left side: \( \frac{3}{x-2} - \frac{x-2}{x-2} < 0 \). This simplifies to \( \frac{3 - (x-2)}{x-2} < 0 \).

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

Determine the critical points by setting the numerator and denominator equal to zero: \( 5 - x = 0 \) gives \( x = 5 \), and \( x - 2 = 0 \) gives \( x = 2 \). These points divide the number line into intervals to test.

Test values from each interval around the critical points \( x = 2 \) and \( x = 5 \) in the inequality \( \frac{5 - x}{x - 2} < 0 \) to determine where the expression is negative. Remember to exclude \( x = 2 \) from the solution set because it makes the denominator zero.

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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 side is a ratio of polynomials. Solving them requires finding values of the variable that make the inequality true, considering where the expression is defined and the sign of the rational expression.

Domain Restrictions

The domain of a rational expression excludes values that make the denominator zero, as division by zero is undefined. Identifying these restrictions is crucial before solving inequalities to avoid invalid solutions.

Interval Notation and Sign Analysis

After determining critical points from the numerator and denominator, the number line is divided into intervals. Testing each interval helps determine where the inequality holds, and the solution is expressed using interval notation to clearly show valid ranges.