Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (x+2)/(2x+3)≤5

Verified step by step guidance

1. Step 1: Start by rewriting the inequality $\frac{x+2}{2x+3}\le 5$ in a form that can be analyzed. Subtract 5 from both sides to get $\frac{x+2}{2x+3}-5\le 0$.
2. Step 2: Combine the terms on the left side over a common denominator: $\frac{x+2-5(2x+3)}{2x+3}\le 0$. Simplify the numerator.
3. Step 3: Simplify the expression in the numerator: $x+2-10x-15=-9x-13$. So the inequality becomes $\frac{-9x-13}{2x+3}\le 0$.
4. Step 4: Identify the critical points by setting the numerator and the denominator equal to zero separately: $-9x-13=0$ and $2x+3=0$. Solve these equations to find the critical points.
5. Step 5: Use the critical points to divide the number line into intervals. Test each interval in the inequality $\frac{-9x-13}{2x+3}\le 0$ to determine where the inequality holds true. Express the solution set in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where a rational function is compared to a value, typically using inequality symbols like ≤, ≥, <, or >. To solve these inequalities, one must find the values of the variable that make the inequality true, often requiring the identification of critical points where the rational expression is zero or undefined.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval (1, 5] includes all numbers greater than 1 and up to 5, including 5 but not 1.

Critical Points

Critical points are values of the variable where the rational expression is either zero or undefined. These points are essential in solving rational inequalities as they divide the number line into intervals. By testing these intervals, one can determine where the inequality holds true, leading to the final solution set.