$$ \left. \begin{array} { l l l } { 31 . - 3 ( v - 3 ) \geq 5 - 4 v } & { 32.3 q + 6 \leq - 5 ( q + 2 ) } & { 33.3 ( 2 + r ) \geq 15 - 2 r } \ { 34.9 + x

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$$ \left. \begin{array} { l l l } { 31 . - 3 ( v - 3 ) \geq 5 - 4 v } & { 32.3 q + 6 \leq - 5 ( q + 2 ) } & { 33.3 ( 2 + r ) \geq 15 - 2 r } \ { 34.9 + x < 7 - 2 ( x - 3 ) } & { 35.2 ( m - 8 ) < - 8 + 3 m } & { 36.2 v - 4 \leq 2 ( 3 v - 6 ) } \end{array} ight. $$

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Certainly! Let's solve each of the given inequalities step by step. --- ### **31.** $$ -3(v - 3) \geq 5 - 4v $$ **Step 1:** Distribute the $$-3$$ on the left side. $$ -3v + 9 \geq 5 - 4v $$ **Step 2:** Add $$4v$$ to both sides to gather like terms. $$ -3v + 4v + 9 \geq 5 $$ $$ v + 9 \geq 5 $$ **Step 3:** Subtract $$9$$ from both sides. $$ v \geq 5 - 9 $$ $$ v \geq -4 $$ **Solution:** $$ v \geq -4 $$ --- ### **32.** $$ 3q + 6 \leq -5(q + 2) $$ **Step 1:** Distribute the $$-5$$ on the right side. $$ 3q + 6 \leq -5q - 10 $$ **Step 2:** Add $$5q$$ to both sides to gather like terms. $$ 3q + 5q + 6 \leq -10 $$ $$ 8q + 6 \leq -10 $$ **Step 3:** Subtract $$6$$ from both sides. $$ 8q \leq -16 $$ **Step 4:** Divide both sides by $$8$$. $$ q \leq -2 $$ **Solution:** $$ q \leq -2 $$ --- ### **33.** $$ 3(2 + r) \geq 15 - 2r $$ **Step 1:** Distribute the $$3$$ on the left side. $$ 6 + 3r \geq 15 - 2r $$ **Step 2:** Add $$2r$$ to both sides to gather like terms. $$ 6 + 5r \geq 15 $$ **Step 3:** Subtract $$6$$ from both sides. $$ 5r \geq 9 $$ **Step 4:** Divide both sides by $$5$$. $$ r \geq \frac{9}{5} \quad \text{or} \quad r \geq 1.8 $$ **Solution:** $$ r \geq \frac{9}{5} $$ or $$ r \geq 1.8 $$ --- ### **34.** $$ 9 + x < 7 - 2(x - 3) $$ **Step 1:** Distribute the $$-2$$ on the right side. $$ 9 + x < 7 - 2x + 6 $$ $$ 9 + x < 13 - 2x $$ **Step 2:** Add $$2x$$ to both sides to gather like terms. $$ 9 + 3x < 13 $$ **Step 3:** Subtract $$9$$ from both sides. $$ 3x < 4 $$ **Step 4:** Divide both sides by $$3$$. $$ x < \frac{4}{3} \quad \text{or} \quad x < 1.\overline{3} $$ **Solution:** $$ x < \frac{4}{3} $$ or $$ x < 1.\overline{3} $$ --- ### **35.** $$ 2(m - 8) < -8 + 3m $$ **Step 1:** Distribute the $$2$$ on the left side. $$ 2m - 16 < -8 + 3m $$ **Step 2:** Subtract $$2m$$ from both sides to gather like terms. $$ -16 < -8 + m $$ **Step 3:** Add $$8$$ to both sides. $$ -8 < m $$ $$ m > -8 $$ **Solution:** $$ m > -8 $$ --- ### **36.** $$ 2v - 4 \leq 2(3v - 6) $$ **Step 1:** Distribute the $$2$$ on the right side. $$ 2v - 4 \leq 6v - 12 $$ **Step 2:** Subtract $$2v$$ from both sides to gather like terms. $$ -4 \leq 4v - 12 $$ **Step 3:** Add $$12$$ to both sides. $$ 8 \leq 4v $$ **Step 4:** Divide both sides by $$4$$. $$ 2 \leq v \quad \text{or} \quad v \geq 2 $$ **Solution:** $$ v \geq 2 $$ --- Feel free to reach out if you have any further questions or need additional explanations! Here are the solutions to the inequalities: 1. $$ v \geq -4 $$ 2. $$ q \leq -2 $$ 3. $$ r \geq \frac{9}{5} $$ or $$ r \geq 1.8 $$ 4. $$ x < \frac{4}{3} $$ or $$ x < 1.\overline{3} $$ 5. $$ m > -8 $$ 6. $$ v \geq 2 $$

\( \left. \begin{array} { l l l } { 31 . - 3 ( v - 3 ) \geq 5 - 4 v } & { 32.3 q + 6 \leq - 5 ( q + 2 ) } & { 33.3 ( 2 + r ) \geq 15 - 2 r } \\ { 34.9 + x < 7 - 2 ( x - 3 ) } & { 35.2 ( m - 8 ) < - 8 + 3 m } & { 36.2 v - 4 \leq 2 ( 3 v - 6 ) } \end{array} \right. \)

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