Question

plotting square roots on the number line
slide the green dot from 0 to plot the number at the correct location.
plot $sqrt{5}$

plot the numbers on the number line.
$sqrt{5}$, $sqrt{8}$, $sqrt{13}$
which inequalities are true? check all that apply.
$square$ $sqrt{5} < 3$
$square$ $sqrt{8} > 3$
$square$ $sqrt{8} < sqrt{13}$ (checked)
$square$ $sqrt{13} = 3.6$
$square$ $2.2 < sqrt{5} < 2.3$
$square$ $3.6 > sqrt{13} > 3.7$

Explanation:

To determine which inequalities are true, we analyze each one:

1. $\boldsymbol{\sqrt{5} < 3}$

Square both sides (both positive, so inequality direction remains): $(\sqrt{5})^2 = 5$, $3^2 = 9$. Since $5 < 9$, $\sqrt{5} < 3$ is true.

2. $\boldsymbol{\sqrt{8} > 3}$

Square both sides: $(\sqrt{8})^2 = 8$, $3^2 = 9$. Since $8 < 9$, $\sqrt{8} < 3$, so $\sqrt{8} > 3$ is false.

3. $\boldsymbol{\sqrt{8} < \sqrt{13}}$

Square both sides: $(\sqrt{8})^2 = 8$, $(\sqrt{13})^2 = 13$. Since $8 < 13$, $\sqrt{8} < \sqrt{13}$ is true (already checked in the image).

4. $\boldsymbol{\sqrt{13} = 3.6}$

Square $3.6$: $3.6^2 = 12.96$. Since $(\sqrt{13})^2 = 13$, and $12.96
eq 13$, $\sqrt{13}
eq 3.6$ (it is approximately $3.6055$), so this is false.

5. $\boldsymbol{2.2 < \sqrt{5} < 2.3}$

Square $2.2$: $2.2^2 = 4.84$; square $2.3$: $2.3^2 = 5.29$. Since $4.84 < 5 < 5.29$, taking square roots (all positive), $2.2 < \sqrt{5} < 2.3$ is true.

6. $\boldsymbol{3.6 > \sqrt{13} > 3.7}$

Square $3.6$: $3.6^2 = 12.96$; square $3.7$: $3.7^2 = 13.69$. Since $12.96 < 13 < 13.69$, taking square roots, $3.6 < \sqrt{13} < 3.7$ (not $3.6 > \sqrt{13} > 3.7$), so this is false.

Final True Inequalities:

• $\sqrt{5} < 3$
• $\sqrt{8} < \sqrt{13}$ (already marked)
• $2.2 < \sqrt{5} < 2.3$

For plotting $\boldsymbol{\sqrt{5}}$:
$\sqrt{5} \approx 2.236$, so move the green dot to a point between $2$ and $3$, closer to $2.2$–$2.3$.

Answer:

(True Inequalities):
$\boldsymbol{\sqrt{5} < 3}$, $\boldsymbol{\sqrt{8} < \sqrt{13}}$, $\boldsymbol{2.2 < \sqrt{5} < 2.3}$ (and the plotted $\sqrt{5}$ is at $\approx 2.236$ on the number line).

(For the checkboxes, mark $\boldsymbol{\sqrt{5} < 3}$, $\boldsymbol{\sqrt{8} < \sqrt{13}}$, and $\boldsymbol{2.2 < \sqrt{5} < 2.3}$ as true.)