Question

graph the solution to the inequality on the number line.

\\|w - 7| \le 4\\

Explanation:

Set up the compound inequality

To solve the absolute value inequality, we rewrite it.
The inequality \(|w - 7| \le 4\) means the distance is at most \(4\).
This translates to:

$$-4 \le w - 7 \le 4$$

Solve for the variable

We isolate \(w\) by adding \(7\) to all parts.

$$-4 + 7 \le w \le 4 + 7$$

$$3 \le w \le 11$$

Determine the graph representation

The solution set is the closed interval \([3, 11]\).
On a number line, this is represented by:

• Solid circles at \(w = 3\) and \(w = 11\) to show inclusion.
• A shaded line segment connecting these two endpoints.

Answer:

The solution to the inequality is \(3 \le w \le 11\).

To graph this on the number line:

• Place a solid circle at \(3\).
• Place a solid circle at \(11\).
• Draw a line segment connecting \(3\) and \(11\).