Question

1. select all of the inequalities that have the solution shown in this number line.$-9leq \frac{x}{4}-8$$\frac{x}{4}+8leq 9$$\frac{x}{4}+8geq 9$$9leq \frac{x}{4}+8$submit

Explanation:

Step1: Analyze the number line

The number line has a closed circle at -4 and the arrow points left, so the solution is $x \leq -4$.

Step2: Solve first inequality $-9 \leq \frac{x}{4} - 8$

Add 8 to both sides:
$-9 + 8 \leq \frac{x}{4}$
$-1 \leq \frac{x}{4}$
Multiply by 4:
$x \geq -4$
This does not match $x \leq -4$.

Step3: Solve second inequality $\frac{x}{4} + 8 \leq 9$

Subtract 8 from both sides:
$\frac{x}{4} \leq 9 - 8$
$\frac{x}{4} \leq 1$
Multiply by 4:
$x \leq 4$
This does not match $x \leq -4$.

Step4: Solve third inequality $\frac{x}{4} + 8 \geq 9$

Subtract 8 from both sides:
$\frac{x}{4} \geq 9 - 8$
$\frac{x}{4} \geq 1$
Multiply by 4:
$x \geq 4$
This does not match $x \leq -4$.

Step5: Solve fourth inequality $9 \leq \frac{x}{4} + 8$

Subtract 8 from both sides:
$9 - 8 \leq \frac{x}{4}$
$1 \leq \frac{x}{4}$
Multiply by 4:
$x \geq 4$
This does not match $x \leq -4$.

Wait, re-check the first inequality (possible sign error):

Step1 (revised): Solve $-9 \leq \frac{x}{4} - 8$

$\frac{x}{4} - 8 \geq -9$
$\frac{x}{4} \geq -9 + 8$
$\frac{x}{4} \geq -1$
$x \geq -4$ (still not matching)

Wait, check if the number line was misread: closed circle at -4, left arrow is $x \leq -4$. Let's test $x=-4$ in each inequality:
1. $-9 \leq \frac{-4}{4} -8 = -1 -8 = -9$ → $-9 \leq -9$ (true, but solution is $x\geq-4$)
2. $\frac{-4}{4} +8 = -1 +8=7 \leq9$ (true, solution $x\leq4$)
3. $\frac{-4}{4} +8=7 \geq9$ (false)
4. $9 \leq \frac{-4}{4} +8=7$ (false)

Wait, maybe the first inequality is $9 \leq \frac{x}{4} -8$? No, as written it's $-9 \leq \frac{x}{4}-8$. Wait, no—wait, let's solve for the correct inequality that gives $x\leq-4$:
Suppose we have $\frac{x}{4} +8 \leq7$ → $\frac{x}{4}\leq-1$ → $x\leq-4$, but that's not an option. Wait, re-express the first inequality:
$-9 \leq \frac{x}{4} -8$ → $\frac{x}{4} \geq -1$ → $x\geq-4$ (this has a closed circle at -4, right arrow, which is opposite of the number line)

Wait, the only inequality that has a solution including $x=-4$ is the first one, but the direction is reversed. Wait, no—wait, maybe the number line is right arrow? No, the arrow points left. Wait, let's recheck each step again:

Step1: Confirm number line solution

Closed dot at -4, left arrow: $x \leq -4$

Step2: Solve $-9 \leq \frac{x}{4} - 8$

$\frac{x}{4} \geq -9 + 8$
$\frac{x}{4} \geq -1$
$x \geq -4$ (solution: right arrow from -4)

Step3: Solve $\frac{x}{4} + 8 \leq 9$

$\frac{x}{4} \leq 1$
$x \leq 4$ (solution: left arrow from 4)

Step4: Solve $\frac{x}{4} + 8 \geq 9$

$\frac{x}{4} \geq 1$
$x \geq 4$ (solution: right arrow from 4)

Step5: Solve $9 \leq \frac{x}{4} + 8$

$\frac{x}{4} \geq 1$
$x \geq 4$ (same as step4)

Wait, there is a discrepancy. But if we assume the number line was misread (right arrow from -4, $x\geq-4$), then the first inequality $-9 \leq \frac{x}{4} - 8$ is correct. But as per the given number line (left arrow from -4), none of the options match? No, wait, maybe a typo in the first inequality: if it was $9 \geq \frac{x}{4} -8$ → $\frac{x}{4} \leq17$ → no. Wait, no, let's test $x=-8$ (which is $\leq-4$) in each inequality:
1. $-9 \leq \frac{-8}{4}-8 = -2-8=-10$ → $-9 \leq -10$ (false)
2. $\frac{-8}{4}+8=-2+8=6 \leq9$ (true, but $x=-8$ is $\leq4$, which is part of $x\leq4$)
3. $\frac{-8}{4}+8=6 \geq9$ (false)
4. $9 \leq 6$ (false)

Ah! The second inequality $\frac{x}{4} + 8 \leq 9$ has solution $x\leq4$, which includes $x\leq-4$ as a subset, but the number line only shows $x\leq-4$, not $x\leq4$. Wait, no, the number line is strictly $x\leq-4$. Wait, maybe the number line's closed dot is at 4, left arrow? No, the dot is at -4.

Wait, re-express the first inequality:
$-9 \leq \frac{x}{4} - 8$
$\frac{x}{4} \geq -1$
$x \geq -4$ (right arrow from -4)

The only inequality that has a solution that includes the number line's shown values (all $x\leq-4$) is the second one, because all $x\leq-4$ are also $x\leq4$. But the number line is specifically $x\leq-4$, not the full $x\leq4$.

Wait, maybe the question has a typo, but based on solving each inequality:
- $-9 \leq \frac{x}{4} - 8$: $x\geq-4$ (right arrow from -4)
- $\frac{x}{4} + 8 \leq 9$: $x\leq4$ (left arrow from 4)
- $\frac{x}{4} + 8 \geq 9$: $x\geq4$ (right arrow from 4)
- $9 \leq \frac{x}{4} + 8$: $x\geq4$ (right arrow from 4)

If the number line was supposed to be right arrow from -4, then the first inequality is correct. If the number line is left arrow from 4, the second is correct. But as per the image, number line is left arrow from -4.

Wait, maybe I misread the first inequality: is it $9 \leq \frac{x}{4} + 8$? No, that's $x\geq4$. Wait, no—wait, let's solve $-9 \geq \frac{x}{4} -8$ (reverse the sign):
$-9 \geq \frac{x}{4} -8$
$\frac{x}{4} \leq -1$
$x \leq -4$, which matches the number line. But the first inequality is written as $-9 \leq \frac{x}{4} -8$, not $\geq$.

Wait, maybe the user's checkmarks are wrong. Based on strict solving:

Answer:

None of the given inequalities match the number line solution $x \leq -4$.
If we assume the number line has a right arrow (solution $x \geq -4$), the correct inequality is: $\boldsymbol{-9 \leq \frac{x}{4} - 8}$
If we assume the number line has a left arrow from 4 (solution $x \leq 4$), the correct inequality is: $\boldsymbol{\frac{x}{4} + 8 \leq 9}$