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Chapter 6: Problem 66

Solve each inequality and graph the solution set on a number line. \(-6 \leq \frac{1}{2} x-4<-3\)

Short Answer

Expert verified
The solution for the compound inequality \(-6 \leq \frac{1}{2} x - 4 < -3\) is \(-4 \leq x < 2\).

Step by step solution

01

Solving the First Inequality

Isolate \(x\) in the first inequality \(-6 \leq \frac{1}{2} x - 4\) by first adding 4 to both sides of the inequality to get rid of the '-4', which gives \(-2 \leq \frac{1}{2} x\). Then multiply each side of the inequality by 2 to solve for \(x\), which yields \(x \geq -4\).
02

Solving the Second Inequality

Next, isolate \(x\) in the second inequality \(\frac{1}{2} x - 4 < -3\). Add 4 to both sides of the inequality to get rid of the '-4', which results in \(\frac{1}{2} x < 1\). Multiplying each side by 2 to isolate \(x\) gives \(x < 2\).
03

Finding the Solution Set

Since the problem is asking for when both inequalities are true, look for where the two solution sets overlap on the number line. The range of \(x\) which is greater than or equal to -4 and less than 2 is the solution set for the compound inequality. Also, use a closed dot at -4 (to represent \(\geq\)) and an open dot at 2 (to represent \(<\)). This yields \(-4 \leq x < 2\).
04

Graphing the Solution

On a number line, plot a closed circle at -4 and an open circle at 2. Draw a line between the two circles to represent the possible values of \(x\) that satisfy both inequalities.

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