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Chapter 0: Problem 56

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

Short Answer

Expert verified
The solution of the compound inequality is \( -4 \leq x < 2 \).

Step by step solution

01

Identify and Isolate the Term

First, isolate the term containing the variable \(x\). In this case, it is \(\frac{1}{2} x - 4\). So, start by adding 4 to all sides of the compound inequality \(-6 \leq \frac{1}{2} x - 4 < -3 \), which results in \(-2 \leq \frac{1}{2} x < 1\).
02

Eliminate the Coefficient of x

Now eliminate the coefficient of \(x\), which is \( \frac{1}{2} \). Multiply through the inequality by \(2\) (the reciprocal of \( \frac{1}{2} \)) to eliminate the fraction: \( -4 \leq x < 2\).
03

Check and Interpret the Solution

At times, it's wise to verify the solution by taking a number from the solution set (for instance, 0) and plug it back into the original inequality to see if it holds. In this case, it does. The solution to this compound inequality is: \( -4 \leq x < 2 \), which can be interpreted as any number equal to or greater than \(-4\) and less than \(2\).

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What is a compound inequality and how is it solved?

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