Answer :
To solve which inequality has open circles on -8 and 2 with a line segment between them, consider the following steps:
### Analyzing the Inequalities
Inequality 1: [tex]\(|x + 8| < 2\)[/tex]
1. The absolute value inequality [tex]\(|x + 8| < 2\)[/tex] can be rewritten as:
[tex]\[ -2 < x + 8 < 2 \][/tex]
2. To solve for [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex] by subtracting 8 from all parts:
[tex]\[ -2 - 8 < x + 8 - 8 < 2 - 8 \][/tex]
[tex]\[ -10 < x < -6 \][/tex]
3. This tells us that [tex]\(x\)[/tex] is between -10 and -6, which means the graph has open circles on -10 and -6 with a line segment between them. This does not match our target condition.
Inequality 2: [tex]\(|x + 3| < 5\)[/tex]
1. The absolute value inequality [tex]\(|x + 3| < 5\)[/tex] can be rewritten as:
[tex]\[ -5 < x + 3 < 5 \][/tex]
2. To solve for [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex] by subtracting 3 from all parts:
[tex]\[ -5 - 3 < x + 3 - 3 < 5 - 3 \][/tex]
[tex]\[ -8 < x < -2 \][/tex]
3. This tells us that [tex]\(x\)[/tex] is between -8 and -2. This set of solutions means the graph has open circles on -8 and -2 with a line segment between them. This does not match our target condition.
Inequality 3: [tex]\(|x + 3| < -5\)[/tex]
1. The inequality [tex]\(|x + 3| < -5\)[/tex] has no solution because the absolute value of any expression is always non-negative, and cannot be less than -5. This inequality is therefore invalid and irrelevant to the problem.
### Conclusion
Only the inequality [tex]\( |x + 8| < 2 \)[/tex] resolves to the solution where the involved points and line segments properly match the conditions stated in the problem.
Therefore, the inequality whose graph has open circles on -8 and 2 with a line segment between them is:
[tex]\[ |x + 8| < 2 \][/tex]
### Analyzing the Inequalities
Inequality 1: [tex]\(|x + 8| < 2\)[/tex]
1. The absolute value inequality [tex]\(|x + 8| < 2\)[/tex] can be rewritten as:
[tex]\[ -2 < x + 8 < 2 \][/tex]
2. To solve for [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex] by subtracting 8 from all parts:
[tex]\[ -2 - 8 < x + 8 - 8 < 2 - 8 \][/tex]
[tex]\[ -10 < x < -6 \][/tex]
3. This tells us that [tex]\(x\)[/tex] is between -10 and -6, which means the graph has open circles on -10 and -6 with a line segment between them. This does not match our target condition.
Inequality 2: [tex]\(|x + 3| < 5\)[/tex]
1. The absolute value inequality [tex]\(|x + 3| < 5\)[/tex] can be rewritten as:
[tex]\[ -5 < x + 3 < 5 \][/tex]
2. To solve for [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex] by subtracting 3 from all parts:
[tex]\[ -5 - 3 < x + 3 - 3 < 5 - 3 \][/tex]
[tex]\[ -8 < x < -2 \][/tex]
3. This tells us that [tex]\(x\)[/tex] is between -8 and -2. This set of solutions means the graph has open circles on -8 and -2 with a line segment between them. This does not match our target condition.
Inequality 3: [tex]\(|x + 3| < -5\)[/tex]
1. The inequality [tex]\(|x + 3| < -5\)[/tex] has no solution because the absolute value of any expression is always non-negative, and cannot be less than -5. This inequality is therefore invalid and irrelevant to the problem.
### Conclusion
Only the inequality [tex]\( |x + 8| < 2 \)[/tex] resolves to the solution where the involved points and line segments properly match the conditions stated in the problem.
Therefore, the inequality whose graph has open circles on -8 and 2 with a line segment between them is:
[tex]\[ |x + 8| < 2 \][/tex]
