Question

solve the following inequality for ( v ).
write your answer in simplest form.

( \begin{align*}
7v + 6 &< 8v - 5\\
-8v quad &quad -8v\\
-v + 6 &< -5\\
quad -6 quad &quad -6\\
-v &< -11
end{align*} )

(switch inequality when dividing by negative)

( \frac{-v}{-1} > \frac{-11}{-1} )

( v > 11 )

Explanation:

Step1: Subtract \(8v\) from both sides

To eliminate the \(8v\) term on the right, we subtract \(8v\) from both sides of the inequality \(7v + 6 < 8v - 5\). This gives \((7v - 8v)+6 < (8v - 8v)-5\), which simplifies to \(-v + 6 < -5\).

Step2: Subtract 6 from both sides

Next, we subtract 6 from both sides to isolate the term with \(v\). So, \(-v + 6 - 6 < -5 - 6\), which simplifies to \(-v < -11\).

Step3: Divide by -1 (and switch inequality)

When we divide both sides of an inequality by a negative number, we must reverse the inequality sign. Dividing both sides of \(-v < -11\) by \(-1\) gives \(\frac{-v}{-1} > \frac{-11}{-1}\), which simplifies to \(v > 11\).

Answer:

\(v > 11\)