Problem 56 - College Algebra Essentials

Question

Solve each compound inequality. $$3 \leq 4 x-3<19$$

Step-by-Step Solution

Answer

The solution to the compound inequality is \(1.5 \leq x < 5.5\).

Step 1: Solve the first inequality

We start by solving the first inequality: \(3 \leq 4x - 3\). To do this, add 3 to both sides of the inequality to isolate the term \(4x\). This gives us \(4x \geq 6\). Then, to finish solving for \(x\), you divide each side by 4 to get \(x \geq 1.5\).

Step 2: Solve the second inequality

Next, solve the second inequality: \(4x - 3 < 19\). As before, add 3 to both sides to isolate \(4x\), yielding \(4x < 22\). Then divide each side by 4 to finish solving for \(x\), leading to \(x < 5.5\).

Step 3: Combine the solutions

The solutions for \(x\) from steps 1 and 2 need to be combined. This results in the final solution for the compound inequality, which is \(1.5 \leq x < 5.5\). This means that \(x\) can be any number between and including 1.5 and excluding 5.5.