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. (3x+5)/(6−2x)≥0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero: $3x+5=0$ and $6-2x=0$.
2. Solve the equations from step 1 to find the critical points: $x=-\frac{5}{3}$ and $x=3$.
3. Determine the intervals to test by using the critical points: $(-\mathrm{\infty },-\frac{5}{3})$, $(-\frac{5}{3},3)$, and $(3,\mathrm{\infty })$.
4. Test a point from each interval in the inequality $\frac{3x+5}{6-2x}\ge 0$ to determine where the inequality holds true.
5. Combine the intervals where the inequality is true and express the solution set in interval notation, considering the critical points where the inequality is equal to zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to zero, typically using symbols like ≥ or ≤. To solve these inequalities, one must determine where the rational expression is positive or negative, which often requires finding critical points where the expression equals zero or is undefined.

Critical Points

Critical points are values of the variable that make the rational expression equal to zero or undefined. These points are essential for determining the intervals on the number line where the inequality holds true. In the given inequality, critical points can be found by solving the equation (3x + 5) = 0 and identifying where the denominator (6 - 2x) is zero.

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 [a, b) includes 'a' but not 'b', which is crucial for expressing the solution set of inequalities accurately.