1. **State the problem:** We have four expressions: $5z^3$, $\frac{3}{z}$, $\sqrt{z}$, and $\frac{z}{5}$. We want to find which inequality comparing two of these expressions is true for $0.7 \leq z \leq 0.8$. 2. **List the inequalities to check:** - $\frac{z}{5} \geq 5z^3$ - $\frac{3}{z} \geq 5z^3$ - $\frac{3}{z} \leq \sqrt{z}$ - $\frac{z}{5} \geq \frac{3}{z}$ 3. **Evaluate each inequality at the endpoints $z=0.7$ and $z=0.8$ to check if it holds throughout the interval:** **Inequality 1: $\frac{z}{5} \geq 5z^3$** - At $z=0.7$: Left $= \frac{0.7}{5} = 0.14$, Right $= 5 \times (0.7)^3 = 5 \times 0.343 = 1.715$; $0.14 \geq 1.715$ is false. - At $z=0.8$: Left $= 0.16$, Right $= 5 \times 0.512 = 2.56$; $0.16 \geq 2.56$ is false. **Inequality 2: $\frac{3}{z} \geq 5z^3$** - At $z=0.7$: Left $= \frac{3}{0.7} \approx 4.286$, Right $= 1.715$; $4.286 \geq 1.715$ is true. - At $z=0.8$: Left $= \frac{3}{0.8} = 3.75$, Right $= 2.56$; $3.75 \geq 2.56$ is true. **Inequality 3: $\frac{3}{z} \leq \sqrt{z}$** - At $z=0.7$: Left $= 4.286$, Right $= \sqrt{0.7} \approx 0.837$; $4.286 \leq 0.837$ is false. - At $z=0.8$: Left $= 3.75$, Right $= \sqrt{0.8} \approx 0.894$; $3.75 \leq 0.894$ is false. **Inequality 4: $\frac{z}{5} \geq \frac{3}{z}$** - At $z=0.7$: Left $= 0.14$, Right $= 4.286$; $0.14 \geq 4.286$ is false. - At $z=0.8$: Left $= 0.16$, Right $= 3.75$; $0.16 \geq 3.75$ is false. 4. **Conclusion:** Only the inequality $\frac{3}{z} \geq 5z^3$ holds true for all $z$ in the interval $0.7 \leq z \leq 0.8$. **Final answer:** $\boxed{\frac{3}{z} \geq 5z^3}$