Solve each rational inequality. Give the solution set in interval notation. 10/(3+2x)≤5

Verified step by step guidance

1. Start by writing down the inequality: \(\frac{10}{3+2x} \leq 5\).
2. Identify the domain restrictions by setting the denominator not equal to zero: \(3 + 2x \neq 0\). Solve for \(x\) to find values to exclude from the solution set.
3. Bring all terms to one side to form a single rational expression: \(\frac{10}{3+2x} - 5 \leq 0\).
4. Combine the terms over a common denominator: \(\frac{10 - 5(3+2x)}{3+2x} \leq 0\). Simplify the numerator.
5. Determine the critical points by setting the numerator and denominator equal to zero separately. Use these points to divide the number line into intervals, then test each interval to see where the inequality holds true, keeping in mind the domain restrictions.

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 where it changes sign.

Domain Restrictions

Since rational expressions have denominators, values that make the denominator zero are excluded from the solution set. Identifying these restrictions is crucial to avoid undefined expressions and to correctly determine solution intervals.

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. Solutions are then expressed in interval notation, clearly showing included and excluded values.