Problem 57 - College Algebra Essentials

Question

Solve each compound inequality. $$-3 \leq \frac{2}{3} x-5<-1$$

Step-by-Step Solution

Answer

The solution to the compound inequality \( -3 \leq \frac{2}{3} x-5<-1 \) is \( 3 \leq x < 6 \).

Step 1: Split the Compound Inequality

The compound inequality -3 ≤ (2/3)x - 5 < -1 can be divided into the two inequalities:\n-3 ≤ (2/3)x - 5 and (2/3)x - 5 < -1

Step 2: Solve the First Inequality

To solve -3 ≤ (2/3)x - 5, first add 5 to both sides of the inequality to get 2 ≤ (2/3)x. Then, multiply both sides of the inequality by 3/2 to get: \(x \geq 3\)

Step 3: Solve the Second Inequality

To solve (2/3)x - 5 < -1, add 5 to both sides of the inequality: (2/3)x < 4. Then, multiply both sides of the inequality by 3/2. This gives: \(x < 6\)

Step 4: Combine the Solutions

Now, combine the solutions from steps 2 and 3 to find the solution set for the original compound inequality. So, the solution is \(3 \leq x < 6\).