$$ 12 x \geq-6 $$ and $$ x-136 $$ $$ 9-3 x+10 \geq 16 $$ or $$ 9-x

Question

$$ 12 x \geq-6 $$ and $$ x-1<0 $$ $$ 2-7 x<14 $$ and $$ 3 x+2 \leq 8 $$ $$ 3-815 $$ or $$ x-5>-6 $$ $$ 84-15 x \geq 4 $$ or $$ 12 x>36 $$ $$ 9-3 x+10 \geq 16 $$ or $$ 9-x<7 $$ $$ 10 \frac{x}{2} \geq-2 $$ and $$ 5-2 x \geq 3 $$

UpStudy AI · Accepted Answer

Let's solve each of the given inequalities step by step. --- ### 1. $$ 12x \geq -6 $$ **and** $$ x - 1 < 0 $$ **Solution:** - **First Inequality:** $$ 12x \geq -6 \implies x \geq \frac{-6}{12} = -\frac{1}{2} $$ - **Second Inequality:** $$ x - 1 < 0 \implies x < 1 $$ **Combined Solution:** $$ -\frac{1}{2} \leq x < 1 $$ --- ### 2. $$ 2 - 7x < 14 $$ **and** $$ 3x + 2 \leq 8 $$ **Solution:** - **First Inequality:** $$ 2 - 7x < 14 \implies -7x < 12 \implies x > -\frac{12}{7} $$ - **Second Inequality:** $$ 3x + 2 \leq 8 \implies 3x \leq 6 \implies x \leq 2 $$ **Combined Solution:** $$ -\frac{12}{7} < x \leq 2 $$ --- ### 3. $$ 3 - 8 < x - 5 < -3 $$ **Solution:** Break it into two parts: 1. $$ 3 - 8 < x - 5 \implies -5 < x - 5 \implies x > 0 $$ 2. $$ x - 5 < -3 \implies x < 2 $$ **Combined Solution:** $$ 0 < x < 2 $$ --- ### 4. $$ 41 \leq 4x + 1 \leq 17 $$ **Solution:** This compound inequality implies: $$ 41 \leq 4x + 1 \quad \text{and} \quad 4x + 1 \leq 17 $$ - From the first part: $$ 4x + 1 \geq 41 \implies 4x \geq 40 \implies x \geq 10 $$ - From the second part: $$ 4x + 1 \leq 17 \implies 4x \leq 16 \implies x \leq 4 $$ **Conclusion:** No solution exists because $$ x \geq 10 $$ and $$ x \leq 4 $$ cannot be true simultaneously. --- ### 5. $$ 51 \leq -2x + 7 < 9 $$ **Solution:** Break it into two parts: 1. $$ 51 \leq -2x + 7 \implies -2x \geq 44 \implies x \leq -22 $$ 2. $$ -2x + 7 < 9 \implies -2x < 2 \implies x > -1 $$ **Conclusion:** No solution exists because $$ x \leq -22 $$ and $$ x > -1 $$ cannot be true simultaneously. --- ### 6. $$ 63x - 1 < -7 $$ **or** $$ 4x + 9 \geq 13 $$ **Solution:** - **First Inequality:** $$ 63x - 1 < -7 \implies 63x < -6 \implies x < -\frac{2}{21} $$ - **Second Inequality:** $$ 4x + 9 \geq 13 \implies 4x \geq 4 \implies x \geq 1 $$ **Combined Solution:** $$ x < -\frac{2}{21} \quad \text{or} \quad x \geq 1 $$ --- ### 7. $$ 7 - 5x > 15 $$ **or** $$ x - 5 > -6 $$ **Solution:** - **First Inequality:** $$ 7 - 5x > 15 \implies -5x > 8 \implies x < -\frac{8}{5} $$ - **Second Inequality:** $$ x - 5 > -6 \implies x > -1 $$ **Combined Solution:** $$ x < -\frac{8}{5} \quad \text{or} \quad x > -1 $$ --- ### 8. $$ 84 - 15x \geq 4 $$ **or** $$ 12x > 36 $$ **Solution:** - **First Inequality:** $$ 84 - 15x \geq 4 \implies -15x \geq -80 \implies x \leq \frac{16}{3} $$ - **Second Inequality:** $$ 12x > 36 \implies x > 3 $$ **Combined Solution:** All real numbers satisfy this inequality because: $$ x \leq \frac{16}{3} \quad \text{or} \quad x > 3 $$ Since $$\frac{16}{3} \approx 5.33 > 3$$, the union covers all real numbers. --- ### 9. $$ 9 - 3x + 10 \geq 16 $$ **or** $$ 9 - x < 7 $$ **Solution:** - **First Inequality:** $$ 19 - 3x \geq 16 \implies -3x \geq -3 \implies x \leq 1 $$ - **Second Inequality:** $$ 9 - x < 7 \implies -x < -2 \implies x > 2 $$ **Combined Solution:** $$ x \leq 1 \quad \text{or} \quad x > 2 $$ --- ### 10. $$ 10 \cdot \frac{x}{2} \geq -2 $$ **and** $$ 5 - 2x \geq 3 $$ **Solution:** - **First Inequality:** $$ 5x \geq -2 \implies x \geq -\frac{2}{5} $$ - **Second Inequality:** $$ 5 - 2x \geq 3 \implies -2x \geq -2 \implies x \leq 1 $$ **Combined Solution:** $$ -\frac{2}{5} \leq x \leq 1 $$ --- ### Summary of All Solutions: 1. **$$-\frac{1}{2} \leq x < 1$$** 2. **$$-\frac{12}{7} < x \leq 2$$** 3. **$$0 < x < 2$$** 4. **No solution** 5. **No solution** 6. **$$x < -\frac{2}{21}$$ or $$x \geq 1$$** 7. **$$x < -\frac{8}{5}$$ or $$x > -1$$** 8. **All real numbers** 9. **$$x \leq 1$$ or $$x > 2$$** 10. **$$-\frac{2}{5} \leq x \leq 1$$** --- If you have any further questions or need additional explanations, feel free to ask! Here are the solutions to the inequalities: 1. $$ -\frac{1}{2} \leq x < 1 $$ 2. $$ -\frac{12}{7} < x \leq 2 $$ 3. $$ 0 < x < 2 $$ 4. No solution 5. No solution 6. $$ x < -\frac{2}{21} $$ or $$ x \geq 1 $$ 7. $$ x < -\frac{8}{5} $$ or $$ x > -1 $$ 8. All real numbers 9. $$ x \leq 1 $$ or $$ x > 2 $$ 10. $$ -\frac{2}{5} \leq x \leq 1 $$

\( 12 x \geq-6 \) and \( x-1<0 \) \( 2-7 x<14 \) and \( 3 x+2 \leq 8 \) \( 3-815 \) or \( x-5>-6 \) \( 84-15 x \geq 4 \) or \( 12 x>36 \) \( 9-3 x+10 \geq 16 \) or \( 9-x<7 \) \( 10 \frac{x}{2} \geq-2 \) and \( 5-2 x \geq 3 \)

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