Question

Solve and graph the inequality.
StartAbsoluteValue x plus 2 EndAbsoluteValue greater than or equals 7
Question content area bottom
Part 1
Choose the correct answer below.
A.x less than or equals minus 9 or x greater than or equals 5
x less than or equals minus 9 or x greater than or equals 5
B.negative 9 less than or equals x less than or equals 5
negative 9 less than or equals x less than or equals 5
C.x greater than or equals 5
x greater than or equals 5
D.x less than or equals minus 7 or x greater than or equals 7
x less than or equals minus 7 or x greater than or equals 7

Answer 1

To solve the inequality \( |x + 2| \geq 7 \), we need to consider the definition of absolute value. The expression \( |A| \geq B \) can be broken down into two separate inequalities:

1. \( A \geq B \)
2. \( A \leq -B \)

For our specific inequality, we will set \( A = x + 2 \) and \( B = 7 \):

### Step 1: Set up the inequalities
From \( |x + 2| \geq 7 \), we can derive:

1. \( x + 2 \geq 7 \)
2. \( x + 2 \leq -7 \)

### Step 2: Solve the inequalities

**For the first inequality:**

\[
x + 2 \geq 7 \\
x \geq 7 - 2 \\
x \geq 5
\]

**For the second inequality:**

\[
x + 2 \leq -7 \\
x \leq -7 - 2 \\
x \leq -9
\]

### Step 3: Combine the results
The solution set can be expressed as:

\[
x \leq -9 \quad \text{or} \quad x \geq 5
\]

### Step 4: Choosing the correct answer
Now, we can compare this with the answer choices:

- **A. \(x \leq -9\) or \(x \geq 5\)** - This is correct.
- **B. \(-9 \leq x \leq 5\)** - This is incorrect.
- **C. \(x \geq 5\)** - This is incomplete.
- **D. \(x \leq -7\) or \(x \geq 7\)** - This is incorrect.

### Conclusion
The correct answer is **A: \(x \leq -9\) or \(x \geq 5\)**.

### Graphing the solution
To graph the solution:

1. Draw a number line.
2. Shade to the left of \( -9 \) (including \( -9 \) with a closed dot).
3. Shade to the right of \( 5 \) (including \( 5 \) with a closed dot).

This reflects the solution set \( (-\infty, -9] \cup [5, \infty) \).