Question

graph the solution to the inequality on the number line. |4v - 4| > 8

Explanation:

Step1: Split the absolute - value inequality

An absolute - value inequality \(|a|>b\) (\(b > 0\)) can be split into two inequalities: \(a>b\) or \(a < - b\). So, for \(|4v - 4|>8\), we have \(4v-4>8\) or \(4v - 4<-8\).

Step2: Solve the first inequality \(4v-4>8\)

Add 4 to both sides of the inequality: \(4v-4 + 4>8 + 4\), which simplifies to \(4v>12\). Then divide both sides by 4: \(\frac{4v}{4}>\frac{12}{4}\), so \(v > 3\).

Step3: Solve the second inequality \(4v-4<-8\)

Add 4 to both sides of the inequality: \(4v-4 + 4<-8 + 4\), which simplifies to \(4v<-4\). Then divide both sides by 4: \(\frac{4v}{4}<\frac{-4}{4}\), so \(v<-1\).

To graph on the number line:

• For \(v > 3\), we use an open - circle at \(v = 3\) and shade to the right.
• For \(v<-1\), we use an open - circle at \(v=-1\) and shade to the left.

Answer:

The solution is \(v<-1\) or \(v > 3\). On the number line, there is an open - circle at \(v=-1\) with shading to the left and an open - circle at \(v = 3\) with shading to the right.