Alright, let's solve the inequality [tex]\( |2x + 4| + 2 > 4 \)[/tex] step-by-step.
### Step 1: Isolate the Absolute Value Expression
First, we need to isolate the absolute value expression:
[tex]\[ |2x + 4| + 2 > 4 \][/tex]
Subtract 2 from both sides:
[tex]\[ |2x + 4| > 2 \][/tex]
### Step 2: Break Down the Absolute Value Inequality
The inequality [tex]\( |2x + 4| > 2 \)[/tex] translates to two separate inequalities:
[tex]\[ 2x + 4 > 2 \quad \text{or} \quad 2x + 4 < -2 \][/tex]
### Step 3: Solve Each Inequality
Now let's solve each inequality separately.
#### Inequality 1: [tex]\( 2x + 4 > 2 \)[/tex]
Subtract 4 from both sides:
[tex]\[ 2x > -2 \][/tex]
Divide by 2:
[tex]\[ x > -1 \][/tex]
#### Inequality 2: [tex]\( 2x + 4 < -2 \)[/tex]
Subtract 4 from both sides:
[tex]\[ 2x < -6 \][/tex]
Divide by 2:
[tex]\[ x < -3 \][/tex]
### Step 4: Combine the Two Solutions
The overall solution to the inequality [tex]\( |2x + 4| > 2 \)[/tex] is the union of the two solutions found in Step 3. This means:
[tex]\[ x > -1 \quad \text{or} \quad x < -3 \][/tex]
In interval notation, the solution is:
[tex]\[ (-\infty, -3) \cup (-1, \infty) \][/tex]
### Final Solution
So the solution to the inequality [tex]\( |2x + 4| + 2 > 4 \)[/tex] is:
[tex]\[ x \in (-\infty, -3) \cup (-1, \infty) \][/tex]
