Problem 58 - College Algebra Essentials

Question

Solve each compound inequality. $$-6 \leq \frac{1}{2} x-4<-3$$

Step-by-Step Solution

Answer

\(-4 \leq x < 2\).

Step 1: Isolate the variable

Start with the inequality \(-6 \leq \frac{1}{2}x - 4\) and add 4 to both sides of the inequality to isolate \(\frac{1}{2}x\). This gives us \(-6 + 4 \leq \frac{1}{2}x\), leading to \(-2 \leq \frac{1}{2}x\). Following the same procedure with the other inequality \(\frac{1}{2}x - 4 < -3\) gives us \(\frac{1}{2}x < -3 + 4\) or \(\frac{1}{2}x < 1\).

Step 2: Solve for x

In both the inequalities derived in Step 1, the variable x is still attached to a factor, \(\frac{1}{2}\). Clear this by multiplying each side of the inequalities by 2. Thus, \(-2 \leq \frac{1}{2}x\) becomes \(-4 \leq x\), and \(\frac{1}{2}x < 1\) is converted to \(x < 2\).

Step 3: Join the inequalites

The isolated inequalities from Step 2, which are \(-4 \leq x\) and \(x < 2\), make up the complete solution to the compound inequality. Combining them gives: \(-4 \leq x < 2\).