1. **State the problem:** Solve the compound inequality $$-6 \leq 2(x - 5) < 7$$. 2. **Understand the inequality:** This is a compound inequality involving a linear expression. We want to find all values of $x$ that satisfy both inequalities simultaneously. 3. **Isolate the variable:** Start by dividing all parts of the inequality by 2 to simplify. $$-6 \leq 2(x - 5) < 7$$ Divide by 2: $$\frac{-6}{\cancel{2}} \leq \frac{2(x - 5)}{\cancel{2}} < \frac{7}{2}$$ which simplifies to $$-3 \leq x - 5 < \frac{7}{2}$$ 4. **Solve for $x$:** Add 5 to all parts of the inequality to isolate $x$. $$-3 + 5 \leq x - 5 + 5 < \frac{7}{2} + 5$$ Simplify: $$2 \leq x < \frac{7}{2} + 5$$ Convert 5 to fraction with denominator 2: $$5 = \frac{10}{2}$$ So, $$x < \frac{7}{2} + \frac{10}{2} = \frac{17}{2}$$ 5. **Final solution:** $$\boxed{2 \leq x < \frac{17}{2}}$$ This means $x$ can be any number greater than or equal to 2 and less than $8.5$.