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Sagot :
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]
### 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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