Question

solve the inequality for v.
\\(\frac{5}{2}v + 1 \leq \frac{3}{2} - \frac{4}{7}v\\)
simplify your answer as much as possible.

Explanation:

Step1: Add $\frac{4}{7}v$ to both sides

To get all $v$ terms on one side, add $\frac{4}{7}v$ to both sides of the inequality $\frac{5}{2}v + 1 \leq \frac{3}{2} - \frac{4}{7}v$. This gives $\frac{5}{2}v + \frac{4}{7}v + 1 \leq \frac{3}{2}$.
First, find a common denominator for the $v$ terms. The common denominator of 2 and 7 is 14. So, $\frac{5}{2}v=\frac{35}{14}v$ and $\frac{4}{7}v=\frac{8}{14}v$. Then, $\frac{35}{14}v+\frac{8}{14}v=\frac{43}{14}v$. So the inequality becomes $\frac{43}{14}v + 1 \leq \frac{3}{2}$.

Step2: Subtract 1 from both sides

Subtract 1 (which is $\frac{2}{2}$) from both sides: $\frac{43}{14}v + 1 - 1 \leq \frac{3}{2} - \frac{2}{2}$. This simplifies to $\frac{43}{14}v \leq \frac{1}{2}$.

Step3: Multiply both sides by $\frac{14}{43}$

To solve for $v$, multiply both sides by the reciprocal of $\frac{43}{14}$, which is $\frac{14}{43}$. So, $v \leq \frac{1}{2} \times \frac{14}{43}$. Simplify the right - hand side: $\frac{1\times14}{2\times43}=\frac{7}{43}$.

Answer:

$v \leq \frac{7}{43}$