Solve each rational inequality. Give the solution set in interval notation. 5/(x+1)>12/(x+1)

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Start by writing down the inequality: $\frac{5}{x+1} > \frac{12}{x+1}$.

Identify the domain restrictions: since the denominators are $x+1$, we must have $x + 1 \neq 0$, so $x \neq -1$.

Because the denominators are the same and not zero, multiply both sides of the inequality by $(x+1)^2$ (which is always positive) to eliminate the denominators without changing the inequality direction. This gives: $5(x+1) > 12(x+1)$.

Simplify the inequality: $5(x+1) > 12(x+1)$ becomes $5x + 5 > 12x + 12$.

Solve the resulting linear inequality for $x$: subtract \$5x$ and \(12$ from both sides to isolate \)x$, then express the solution set in interval notation, remembering to exclude $x = -1$ from the domain.

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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 where variables appear 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 crucial to identify values that make denominators zero, as these values are excluded from the solution set. These restrictions define the domain and help avoid invalid solutions.

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 essential.