Question

1. -8(b - 2) - 8b < -112

Explanation:

Step1: Distribute -8

First, we distribute the -8 in the term \(-8(b - 2)\). Using the distributive property \(a(b - c)=ab - ac\), we get:
\(-8(b - 2)=-8b+16\)
So the inequality becomes \(-8b + 16-8b<-112\)

Step2: Combine like terms

Next, we combine the like terms \(-8b\) and \(-8b\). Combining these gives:
\(-8b-8b=-16b\)
So the inequality is now \(-16b + 16<-112\)

Step3: Subtract 16 from both sides

We subtract 16 from both sides of the inequality to isolate the term with \(b\).
\(-16b+16 - 16<-112 - 16\)
Simplifying both sides, we have:
\(-16b<-128\)

Step4: Divide by -16 (reverse inequality)

When we divide both sides of an inequality by a negative number, we must reverse the inequality sign. So we divide both sides by \(-16\):
\(\frac{-16b}{-16}>\frac{-128}{-16}\)
Simplifying, we get:
\(b > 8\)

Answer:

The solution to the inequality \(-8(b - 2)-8b<-112\) is \(b>8\). On the number line, this would be represented by an open circle at 8 and an arrow pointing to the right (towards values greater than 8).