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 + 1)/(x + 3) < 2

검증된 단계별 안내

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

Find a common denominator to combine the terms into a single rational expression: write \(2\) as \(\frac{2(x + 3)}{x + 3}\), so the inequality becomes \(\frac{x + 1}{x + 3} - \frac{2(x + 3)}{x + 3} < 0\).

Combine the fractions into one rational expression: \(\frac{x + 1 - 2(x + 3)}{x + 3} < 0\). Simplify the numerator by distributing and combining like terms.

Identify the critical points by setting the numerator and denominator equal to zero separately. These points divide the number line into intervals to test for the inequality.

Test each interval determined by the critical points in the simplified rational expression to determine where the expression is less than zero. Use these results to write the solution set in interval notation and graph it on the real number line.

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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 rational expression is compared to another using inequality symbols. Solving them requires finding values of the variable that make the inequality true, often by bringing all terms to one side and analyzing the sign of the resulting expression.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator of a rational expression equals zero, causing the expression to change sign or be undefined. Identifying these points allows you to divide the number line into intervals and test each interval to determine where the inequality holds.

Interval Notation and Graphing Solutions

Interval notation is a concise way to represent solution sets on the real number line, using parentheses or brackets to indicate whether endpoints are included. Graphing the solution helps visualize the intervals where the inequality is true, clarifying the solution set.