Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x + 4)/(2x - 1) ≤ 3

Verified step by step guidance

1. Step 1: Start by moving all terms to one side of the inequality to set it to zero: $\frac{x+4}{2x-1}-3\le 0$.
2. Step 2: Combine the terms into a single rational expression: $\frac{x+4-3(2x-1)}{2x-1}\le 0$.
3. Step 3: Simplify the numerator: $x+4-6x+3=-5x+7$. The inequality becomes $\frac{-5x+7}{2x-1}\le 0$.
4. Step 4: Identify the critical points by setting the numerator and denominator equal to zero: $-5x+7=0$ and $2x-1=0$. 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 $\frac{-5x+7}{2x-1}\le 0$ to determine where the inequality holds true. Express the solution set in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation 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 expression equals zero or is undefined.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2.

Graphing Solution Sets

Graphing solution sets involves visually representing the solutions of an inequality on a number line. This includes marking critical points, shading the appropriate regions to indicate where the inequality holds true, and using open or closed circles to denote whether endpoints are included in the solution set. This visual representation aids in understanding the range of solutions.