The graph in the image shows a number line with points marked at -8, -2, and a line segment connecting these points. This suggests the inequality involves absolute value.

The given options are:

1. $|x + 5| < 3$
2. $|x + 5| \leq 3$
3. $|x + 5| > 3$
4. $|x + 5| \geq 3$

Analyzing the Inequality

To determine the correct inequality, let's consider each option and what it represents on a number line:

1. $|x + 5| < 3$:

   • This means the distance from $-5$ is less than $3$.
   • The solution would be: $-8 < x + 5 < -2$.
   • Subtract $5$ from all parts: $-13 < x < -7$.
   • This doesn't match the segment between $-8$ and $-2$.

2. $|x + 5| \leq 3$:

   • This means the distance from $-5$ is less than or equal to $3$.
   • The solution would be: $-8 \leq x + 5 \leq -2$.
   • Subtract $5$ from all parts: $-13 \leq x \leq -7$.
   • This also doesn't match the segment between $-8$ and $-2$.

3. $|x + 5| > 3$:

   • This means the distance from $-5$ is more than $3$.
   • The solution would be: $x + 5 > -2$ or $x + 5 < -8$.
   • Subtract $5$ from all parts: $x > -7$ or $x < -13$.
   • This doesn't match the segment between $-8$ and $-2$.

4. $|x + 5| \geq 3$:

   • This means the distance from $-5$ is more than or equal to $3$.
   • The solution would be: $x + 5 \geq -2$ or $x + 5 \leq -8$.
   • Subtract $5$ from all parts: $x \geq -7$ or $x \leq -13$.
   • This also doesn't match the segment between $-8$ and $-2$.

Correct Inequality

The inequality that matches the given graph is:

$|x + 5| \leq 3$

Explanation:

• The graph shows a closed interval $[-8, -2]$.
• This represents all $x$ values such that the distance from $-5$ is less than or equal to $3$.
• The correct inequality is $|x + 5| \leq 3$.

Would you like further details or have any questions?

Relative Questions:

1. What is the general solution for the inequality $|x + a| \leq b$?
2. How do you graph $|x - a| \leq b$ on a number line?
3. Can you explain why $|x + 5| \leq 3$ and $|x + 5| < 3$ have different solution sets?
4. How would the graph change if the inequality were $|x + 5| > 3$?
5. What are the steps to solve absolute value inequalities?

Tip:

When solving absolute value inequalities, remember to consider both the positive and negative scenarios to find the complete solution set.