1. **Problem Statement:** Leila is designing a rectangular table with an area of 12 square feet or less. The length is given as $\frac{3x}{4}$ and the width is $x$. We want to find the range of values for $x$. 2. **Formula for Area of a Rectangle:** $$\text{Area} = \text{length} \times \text{width}$$ Here, area $A = \frac{3x}{4} \times x = \frac{3x^2}{4}$. 3. **Inequality Setup:** Since the area must be 12 square feet or less, we write: $$\frac{3x^2}{4} \leq 12$$ 4. **Solve the Inequality:** Multiply both sides by 4 to eliminate the denominator: $$3x^2 \leq 48$$ Divide both sides by 3: $$\cancel{3}x^2 \leq \cancel{3}16$$ $$x^2 \leq 16$$ 5. **Find the range for $x$:** Take the square root of both sides: $$-4 \leq x \leq 4$$ 6. **Contextual Reasoning:** Since $x$ represents a side length, it must be positive: $$0 < x \leq 4$$ 7. **Final Answer:** The range of values for $x$ is: $$0 < x \leq 4$$ --- **Additional Problems:** **Problem 2:** Solve $-\frac{2x}{5} \leq 2$. Multiply both sides by $-\frac{5}{2}$ and reverse inequality: $$x \geq -5$$ **Problem 3:** Solve $x - 2 > -6$. Add 2 to both sides: $$x > -4$$ **Problem 4 & 5:** Haru's window with width 4 ft and area no more than 24 sq ft. Inequality: $$4x \leq 24$$ Divide both sides by 4: $$x \leq 6$$ **Problem 6:** Solve $-\frac{3}{2}x < 18$. Multiply both sides by $-\frac{2}{3}$ and reverse inequality: $$x > -12$$ **Summary:** - When multiplying or dividing by a negative number, reverse the inequality sign. - Side lengths must be positive in real-world contexts. - Inequalities describe ranges of possible values.