Solve each rational inequality. Give the solution set in interval notation. 7/(x+2)≥1/(x+2)

검증된 단계별 안내

Start by writing down the inequality: \(\frac{7}{x+2} \geq \frac{1}{x+2}\).

Identify the domain restrictions: since the denominators contain \(x+2\), set \(x+2 \neq 0\), so \(x \neq -2\) to avoid division by zero.

Bring all terms to one side to compare them: \(\frac{7}{x+2} - \frac{1}{x+2} \geq 0\).

Combine the fractions since they have the same denominator: \(\frac{7 - 1}{x+2} \geq 0\), which simplifies to \(\frac{6}{x+2} \geq 0\).

Analyze the inequality \(\frac{6}{x+2} \geq 0\): since the numerator 6 is positive, the sign of the expression depends on the denominator \(x+2\). Determine where \(x+2 > 0\) and where \(x+2 < 0\), then write the solution set accordingly, remembering to exclude \(x = -2\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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.

Domain Restrictions

When solving rational inequalities, it is essential to identify values that make any denominator zero, as these values are excluded from the solution set. This ensures the solution only includes valid inputs where the expression is defined.

Interval Notation and Solution Sets

After solving the inequality, the solution set is expressed in interval notation, which concisely represents all values satisfying the inequality. Understanding how to write and interpret intervals, including open and closed endpoints, is crucial.