Question

solve the inequality for v. 7v - 32 ≤ -3(6 - 3v) simplify your answer as much as possible.

Explanation:

Step1: Expand the right - hand side

We have the inequality \(7v - 32\leq- 3(6 - 3v)\). First, use the distributive property \(a(b - c)=ab - ac\) (here \(a=-3\), \(b = 6\), \(c = 3v\)) to expand the right - hand side:
\(-3(6 - 3v)=-3\times6-(-3)\times3v=-18 + 9v\)
So the inequality becomes \(7v-32\leq - 18 + 9v\)

Step2: Move the terms with v to one side

Subtract \(7v\) from both sides of the inequality:
\(7v-32-7v\leq - 18 + 9v-7v\)
Simplifying both sides, we get \(-32\leq - 18+2v\)

Step3: Isolate the term with v

Add 18 to both sides of the inequality:
\(-32 + 18\leq-18 + 2v+18\)
Simplifying both sides, \(-14\leq2v\)

Step4: Solve for v

Divide both sides of the inequality by 2:
\(\frac{-14}{2}\leq\frac{2v}{2}\)
Simplifying, we get \(-7\leq v\) or \(v\geq - 7\)

Answer:

\(v\geq - 7\)