Solve each rational inequality. Give the solution set in interval notation. 10/(x+3)≥1

Verified step by step guidance

1. Start by rewriting the inequality: \(\frac{10}{x+3} \geq 1\).
2. Bring all terms to one side to have zero on the other side: \(\frac{10}{x+3} - 1 \geq 0\).
3. Find a common denominator and combine the terms: \(\frac{10 - (x+3)}{x+3} \geq 0\), which simplifies to \(\frac{7 - x}{x+3} \geq 0\).
4. Determine the critical points by setting numerator and denominator equal to zero: numerator \(7 - x = 0\) gives \(x = 7\), denominator \(x + 3 = 0\) gives \(x = -3\) (excluded from domain).
5. Use these critical points to divide the number line into intervals and test each interval in the inequality \(\frac{7 - x}{x+3} \geq 0\) to find where the expression is nonnegative, then express the solution set in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one side is a ratio of polynomials. Solving them requires finding values of the variable that make the inequality true, considering where the expression is defined and the sign of the numerator and denominator.

Domain Restrictions

The domain of a rational expression excludes values that make the denominator zero, as division by zero is undefined. Identifying these restrictions is crucial before solving inequalities to avoid invalid solutions.

Interval Notation and Sign Analysis

After finding critical points from numerator and denominator, the number line is divided into intervals. Testing each interval determines where the inequality holds. The solution is then expressed in interval notation, showing all valid values.