Practice Solving Multi-Step Inequalities

Reduce any fractions to lowest terms. Don't round your answer, and don't use mixed fractions.

1. Solve for \(v\).
   

   \(-42v+33 < 8v + 91\)

2. Solve for \(q\).
   

   \(50q+43 > -11q + 70\)

3. Solve for \(y\).
   

   \(-65y+19 < -2y + 41\)

4. Solve for \(t\).
   

   \(-48t+2 \leq -71t + 14\)

Source: Khan Academy, https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:solve-equations-inequalities/x2f8bb11595b61c86:multistep-inequalities/e/linear_inequalities
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1. 
   \(\begin{aligned}-42v+33 & < 8v + 91 \\\\
   -42v& < 8v+58 &(\text{Subtract } 33 \text{ from both sides}) \\\\
   -50v & < 58 &(\text{Subtract } 8v \text{ from both sides})\\\\
   50v& > -58&(\text{Multiply both sides by }-1)\\\\
   v& > -\dfrac{29}{25}&(\text{Divide both sides by }50 \text{ and simplify})
   \end{aligned}\)

   In conclusion, the answer is \(v > - \dfrac{29}{25}\).
   

2. 
   \(\begin{aligned}50q+43 & > -11q + 70 \\\\
   50q& > -11q+27 &(\text{Subtract } 43 \text{ from both sides}) \\\\
   61q & > 27 &(\text{Add } 11q \text{ to both sides})\\\\
   q& > \dfrac{27}{61}&(\text{Divide both sides by }61
   ) \end{aligned}\)

   In conclusion, the answer is \(q > \dfrac{27}{61}\).

3. 
   \(\begin{aligned}-65y+19 & < -2y + 41 \\\\
   -65y& < -2y+22 &(\text{Subtract } 19 \text{ from both sides}) \\\\
   -63y & < 22 &(\text{Add } 2y \text{ to both sides})\\\\
   63y& > -22&(\text{Multiply both sides by }-1)\\\\
   y& > -\dfrac{22}{63}&(\text{Divide both sides by }63)
   \end{aligned}\)

   In conclusion, the answer is \(y > - \dfrac{22}{63}\).

4. 
   \(\begin{aligned}-48t+2 & \leq -71t + 14 \\\\
   -48t&\leq -71t+12 &(\text{Subtract } 2 \text{ from both sides}) \\\\
   23t &\leq 12 &(\text{Add } 71t \text{ to both sides})\\\\
   t&\leq\dfrac{12}{23}&(\text{Divide both sides by }23)
    \end{aligned}\)

   In conclusion, the answer is \(t \leq \dfrac{12}{23}\).