Question

Use the image to answer the question.

A number line ranges from negative 10 to 10 in increments of 1. Two closed points are plotted at negative 9 and 9, with a solid line extending between them.

Which inequality’s solution set does the graph represent?

(1 point)
Responses

|x|+5≥9
Start Absolute Value x End Absolute Value plus 5 greater-than-or-equal-to 9

12|x+3|≤−4
Start Fraction 1 over 2 End Fraction Start Absolute Value x plus 3 End Absolute Value less-than-or-equal-to negative 4

|x|−4≤5
Start Absolute Value x End Absolute Value minus 4 less-than-or-equal-to 5

2|x−8|≤14

Answer 1

To determine which inequality’s solution set the graph represents, let's analyze the plotted points and the solid line on the number line. The solid line extends from \(-9\) to \(9\), which means the values of \(x\) are within this range, inclusive. Therefore, the solution is represented by the inequality:

\(-9 \leq x \leq 9\)

Now we will check each option to see which one corresponds to this solution set.

1. **|x| + 5 ≥ 9**
Simplifying gives |x| ≥ 4, which leads to x ≤ -4 or x ≥ 4. This does not match the solution.

2. **12|x + 3| ≤ -4**
Since the left side, 12|x + 3|, is always non-negative, this inequality cannot be true because it states that a non-negative quantity is less than or equal to a negative number. Thus, this does not represent a valid solution.

3. **|x| - 4 ≤ 5**
Simplifying gives |x| ≤ 9. The solution set will be from -9 to 9, which aligns with the number line (i.e., -9 ≤ x ≤ 9). This matches our solution.

4. **2|x - 8| ≤ 14**
Dividing by 2 gives |x - 8| ≤ 7. This leads to 8 - 7 ≤ x ≤ 8 + 7, or 1 ≤ x ≤ 15. This does not match our range.

Therefore, the correct answer, representing the solution set depicted in the graph, is:

**|x| - 4 ≤ 5**