Question

linear inequalities (level 2)
question
solve the following inequality for v. write your answer in simplest form.
-4v - 2(3v + 9) ≤ 9v - 2 + v

Explanation:

Step1: Expand the left side

First, we expand \(-2(3v + 9)\) using the distributive property \(a(b + c)=ab+ac\). So, \(-2(3v + 9)=-6v-18\). Then the left side of the inequality becomes \(-4v-6v - 18=-10v-18\).

Step2: Combine like terms on the right side

On the right side, we combine \(9v\) and \(v\) to get \(10v\). So the inequality is now \(-10v-18\leq10v - 2\).

Step3: Add \(10v\) to both sides

Adding \(10v\) to both sides to get all \(v\) terms on one side: \(-10v+10v - 18\leq10v+10v - 2\), which simplifies to \(-18\leq20v - 2\).

Step4: Add 2 to both sides

Adding 2 to both sides: \(-18 + 2\leq20v-2 + 2\), which simplifies to \(-16\leq20v\).

Step5: Divide both sides by 20

Divide both sides by 20: \(\frac{-16}{20}\leq\frac{20v}{20}\). Simplify \(\frac{-16}{20}\) to \(\frac{-4}{5}\), so we get \(-\frac{4}{5}\leq v\) or \(v\geq-\frac{4}{5}\).

Answer:

\(v\geq-\frac{4}{5}\)