10. Solve the inequalities and compare. a. Solve $$ 2 x+6

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10. Solve the inequalities and compare. a. Solve $$ 2 x+6<10 $$. b. Solve $$ -2 x+22<18 $$. c. Which is the correct comparison of solutions for $$ 2 x+6<10 $$ and $$ -2 x+22<18 $$ ? (A) The inequalities have some common solutions. (B) The inequalities have one common solution. (C) The inequalities have no common solutions. (D) The inequalities have the same solutions.

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**Step 1. Solve $$2x+6<10$$** Subtract $$6$$ from both sides: $$ 2x < 10 - 6 \quad \Longrightarrow \quad 2x < 4 $$ Divide both sides by $$2$$: $$ x < \frac{4}{2} \quad \Longrightarrow \quad x < 2 $$ **Step 2. Solve $$-2x+22<18$$** Subtract $$22$$ from both sides: $$ -2x < 18 - 22 \quad \Longrightarrow \quad -2x < -4 $$ Divide both sides by $$-2$$ (remember to reverse the inequality sign): $$ x > \frac{-4}{-2} \quad \Longrightarrow \quad x > 2 $$ **Step 3. Compare the Solutions** The solution for $$2x+6<10$$ is: $$ x < 2 $$ The solution for $$-2x+22<18$$ is: $$ x > 2 $$ Since there is no value of $$x$$ that is simultaneously less than $$2$$ and greater than $$2$$, the inequalities have no common solutions. **Answer:** (C) The inequalities have no common solutions. The inequalities $$2x+6<10$$ and $$-2x+22<18$$ have no common solutions.

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