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

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Start by writing the inequality clearly: \(\frac{4}{x+1} < \frac{2}{x+3}\).

Bring all terms to one side to have zero on the other side: \(\frac{4}{x+1} - \frac{2}{x+3} < 0\).

Find a common denominator, which is \((x+1)(x+3)\), and combine the fractions: \(\frac{4(x+3) - 2(x+1)}{(x+1)(x+3)} < 0\).

Simplify the numerator: \(4(x+3) - 2(x+1) = 4x + 12 - 2x - 2 = 2x + 10\), so the inequality becomes \(\frac{2x + 10}{(x+1)(x+3)} < 0\).

Determine the critical points by setting numerator and denominator equal to zero: numerator \(2x + 10 = 0\) gives \(x = -5\), denominator factors \(x+1=0\) and \(x+3=0\) give \(x = -1\) and \(x = -3\). Use these points to test intervals on the number line to find where the expression is less than zero.

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주요 개념

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

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. Solving them requires finding values of the variable that make the inequality true, while considering restrictions where the denominator is zero to avoid undefined expressions.

Finding a Common Denominator and Combining Terms

To solve rational inequalities, it is often necessary to rewrite both sides with a common denominator. This allows combining the inequality into a single rational expression, making it easier to analyze the sign of the expression.

Sign Analysis and Interval Testing

After combining terms, determine where the numerator and denominator are zero to find critical points. Use these points to divide the number line into intervals and test each interval to see where the inequality holds, considering domain restrictions.