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Solve the inequality for [tex]$x$[/tex] and identify the graph of its solution.

[tex] |x+2|\ \textgreater \ 2 [/tex]

Choose the answer that gives both the correct solution and the correct graph.

A. Solution: [tex]x\ \textgreater \ 0[/tex] and [tex]x\ \textless \ 4[/tex]

B. Solution: [tex]x\ \textless \ -4[/tex] or [tex]x\ \textgreater \ 0[/tex]

C. Solution: [tex]x\ \textgreater \ -4[/tex] and [tex]x\ \textless \ 0[/tex]

D. Solution: [tex]x\ \textless \ -4[/tex] or [tex]x\ \textgreater \ 0[/tex]

Sagot :

GinnyAnswer
Let's solve the inequality [tex]\( |x+2| > 2 \)[/tex] step by step.

### Step 1: Understand the Absolute Value Inequality
The inequality [tex]\( |x+2| > 2 \)[/tex] can be broken down into two separate inequalities:
1. [tex]\( x + 2 > 2 \)[/tex]
2. [tex]\( x + 2 < -2 \)[/tex]

### Step 2: Solve Each Inequality Separately
#### Solve [tex]\( x + 2 > 2 \)[/tex]
[tex]\[ x + 2 > 2 \][/tex]
Subtract 2 from both sides:
[tex]\[ x > 0 \][/tex]

#### Solve [tex]\( x + 2 < -2 \)[/tex]
[tex]\[ x + 2 < -2 \][/tex]
Subtract 2 from both sides:
[tex]\[ x < -4 \][/tex]

### Step 3: Combine the Solutions
The solutions from both inequalities are [tex]\( x > 0 \)[/tex] and [tex]\( x < -4 \)[/tex].

### Step 4: Interpret the Combined Solution
The inequality [tex]\( |x+2| > 2 \)[/tex] is satisfied when [tex]\( x \)[/tex] is either greater than 0 or less than -4.

### Step 5: Choose the Correct Answer and Graph
The correct solution range is [tex]\( x > 0 \)[/tex] or [tex]\( x < -4 \)[/tex].

Checking the options:
- Option D: Solution: [tex]\( x < -4 \)[/tex] or [tex]\( x > 0 \)[/tex]

Thus, the correct answer is:
D. Solution: [tex]\( x < -4 \)[/tex] or [tex]\( x > 0 \)[/tex]
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