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

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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 $x = -2$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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