Problem 50 - Algebra and Trigonometry

Question

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. $$ \frac{3 x+5}{6-2 x} \geq 0 $$

Step-by-Step Solution

Answer

The solution to the inequality in interval notation is \( [-5/3, 3) \)

Step 1: Identify Critical Points

Critical points are obtained as the solutions for the equations \( 3x + 5 = 0 \) (numerator) and \( 6 - 2x = 0 \) (denominator). To solve these equations, subtract 5 and add \( 2x \) respectively, giving \( x = -5/3 \) and \( x = 3 \).

Step 2: Create a Number Line

Plot the critical points on the number line. They divide the number line into three intervals: \( (-\infty, -5/3) \), \( (-5/3, 3) \), and \( (3, +\infty) \).

Step 3: Test Intervals

Choose a number from each interval and substitute it into the original inequality. For the first interval, let's test \( x = -2 \). Thus \( \frac{3(-2)+5}{6-2(-2)} = -1/2 \) which is less than 0. This implies all numbers in this interval are not solutions to the inequality. For the second interval, let's choose \( x = 0 \). The inequality becomes \( \frac{3(0)+5}{6-2(0)} = 5/6 \) which is greater than 0, meaning that all numbers in this interval are solutions. Finally, for the interval \( x > 3 \), choose \( x = 4 \). This gives \( \frac{3(4)+5}{6-2(4)} = -7 \), which does not satisfy the inequality.

Step 4: Express the Solution in Interval Notation

From the above analysis, the solution to the inequality \( \frac{3 x+5}{6-2 x} \geq 0 \) in interval notation is \( [-5/3, 3) \)