1. **Problem A:** Determine the truth of the biconditional "if $\sqrt{x} \geq x$ if and only if $0 < x < 1$." - The functions involved are $y = \sqrt{x}$ and $y = x$. - We want to check where $\sqrt{x} \geq x$ holds true. 2. **Formula and rules:** - Recall that $\sqrt{x}$ is defined for $x \geq 0$. - To compare $\sqrt{x}$ and $x$, consider the inequality: $$\sqrt{x} \geq x$$ - Square both sides (valid since both sides are non-negative for $x \geq 0$): $$x \geq x^2$$ 3. **Intermediate work:** - Rewrite as: $$x - x^2 \geq 0$$ - Factor: $$x(1 - x) \geq 0$$ 4. **Analyze the inequality:** - The product $x(1-x)$ is non-negative when $x \in [0,1]$. - At $x=0$ or $x=1$, the expression equals zero. - For $0 < x < 1$, $\sqrt{x} > x$. - For $x > 1$, $\sqrt{x} < x$. 5. **Conclusion for A:** - The biconditional "$\sqrt{x} \geq x$ if and only if $0 < x < 1$" is true if we exclude $x=0$. - Including $x=0$ makes the inequality true as well, so the biconditional is true for $0 \leq x \leq 1$. --- 6. **Problem B:** Determine the truth of the biconditional "$x^2 < x^3$ if and only if $x > 0$." 7. **Formula and rules:** - Consider the inequality: $$x^2 < x^3$$ - For $x \neq 0$, divide both sides by $x^2$: $$\cancel{x^2} < \cancel{x^2} x \Rightarrow 1 < x$$ 8. **Analyze the inequality:** - The inequality $x^2 < x^3$ holds when $x > 1$. - For $0 < x < 1$, $x^3 < x^2$, so inequality does not hold. 9. **Conclusion for B:** - The biconditional "$x^2 < x^3$ if and only if $x > 0$" is false. - The correct biconditional is "$x^2 < x^3$ if and only if $x > 1$." --- 10. **Problem C:** Biconditional "A whole number is divisible by 3 if and only if it is divisible by 6." 11. **Analyze divisibility:** - If a number is divisible by 6, it is divisible by 3 (true). - If a number is divisible by 3, it is not necessarily divisible by 6 (false, e.g., 3). 12. **Conclusion for C:** - The biconditional is false because one direction fails. --- 13. **Problem D:** Biconditional "A whole number is even if and only if it is divisible by 2." 14. **Analyze definitions:** - By definition, even numbers are divisible by 2. - If a number is divisible by 2, it is even. 15. **Conclusion for D:** - The biconditional is true. --- **Final summary:** - A: True biconditional for $0 \leq x \leq 1$. - B: False biconditional; correct is $x > 1$. - C: False biconditional. - D: True biconditional.