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. (4−2x)/(3x+4)≤0

Verified step by step guidance

1. Identify the rational inequality: \(\frac{4 - 2x}{3x + 4} \leq 0\).
2. Find the critical points by setting the numerator and denominator equal to zero separately: solve \(4 - 2x = 0\) and \(3x + 4 = 0\).
3. Determine the values of \(x\) where the expression is undefined (denominator zero) and where the expression equals zero (numerator zero). These points divide the number line into intervals.
4. Test a value from each interval in the inequality \(\frac{4 - 2x}{3x + 4} \leq 0\) to check whether the expression is negative or zero in that interval.
5. Combine the intervals where the inequality holds true, and express the solution set in interval notation, remembering to exclude points where the denominator is zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the expression is positive, negative, or zero by analyzing the signs of numerator and denominator.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, dividing the number line into intervals. By testing values in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation and Graphing Solutions

After determining solution intervals, express them using interval notation, which concisely represents sets of real numbers. Graphing on a number line visually shows these intervals, indicating included or excluded endpoints based on inequality type.