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Question

Part 1
Solve the inequality. Graph the solution.
StartAbsoluteValue x plus 2 EndAbsoluteValue greater than or equals 7x+2≥7
Question content area bottom
Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) to complete your choice.

Answer 1

To solve the inequality \(|x + 2| \geq 7\), we will break it down into two cases based on the properties of absolute values.

Case 1: \(x + 2 \geq 7\)

1. Solve the inequality: \[ x + 2 \geq 7 \] \[ x \geq 7 - 2 \] \[ x \geq 5 \]

Case 2: \(x + 2 \leq -7\)

1. Solve the inequality: \[ x + 2 \leq -7 \] \[ x \leq -7 - 2 \] \[ x \leq -9 \]

Combine the Solutions

From the two cases, we have:

1. \(x \geq 5\)
2. \(x \leq -9\)

Solution in Interval Notation

The complete solution set can be written in interval notation as: \[ (-\infty, -9] \cup [5, \infty) \]

Graphing the Solution

To graph the solution:

• Draw a number line.
• Shade the region to the left of -9 (including -9, which is a closed circle).
• Shade the region to the right of 5 (including 5, which is also a closed circle).

Select the Correct Choice

The final answer is: \[ x \leq -9 \quad \text{or} \quad x \geq 5 \] Express your answer in interval notation: \[ (-\infty, -9] \cup [5, \infty) \]