To solve the absolute value inequality [tex]\( |2x + 4| > 14 \)[/tex], we will first break it down into two separate inequalities, since the absolute value of a number is greater than some positive value if and only if that number is either greater than that value or less than the negative of that value.
So, we consider:
[tex]\[ 2x + 4 > 14 \][/tex]
and
[tex]\[ 2x + 4 < -14 \][/tex]
Step 1: Solve the first inequality
[tex]\[ 2x + 4 > 14 \][/tex]
Subtract 4 from both sides:
[tex]\[ 2x > 10 \][/tex]
Divide both sides by 2:
[tex]\[ x > 5 \][/tex]
Step 2: Solve the second inequality
[tex]\[ 2x + 4 < -14 \][/tex]
Subtract 4 from both sides:
[tex]\[ 2x < -18 \][/tex]
Divide both sides by 2:
[tex]\[ x < -9 \][/tex]
Now we have two intervals:
[tex]\[ x > 5 \][/tex]
and
[tex]\[ x < -9 \][/tex]
### Solution in interval notation
The solution to the inequality [tex]\( |2x + 4| > 14 \)[/tex] is:
[tex]\[ x \in (-\infty, -9) \cup (5, \infty) \][/tex]
### Graphing the solution
To graph the solution on a number line:
1. Draw a number line.
2. Mark the points [tex]\( -9 \)[/tex] and [tex]\( 5 \)[/tex] on the number line.
3. Shade the region to the left of [tex]\( -9 \)[/tex] (including all values less than [tex]\( -9 \)[/tex]).
4. Shade the region to the right of [tex]\( 5 \)[/tex] (including all values greater than [tex]\( 5 \)[/tex]).
5. Use open circles at [tex]\( -9 \)[/tex] and [tex]\( 5 \)[/tex] to indicate that these points are not included in the solution set.
Here is a rough illustration of the number line:
```
<---|-------|-------------|-------|-------------|--->
-∞ -9 0 5 ∞
( )
(————— —————)
```
The graph shows that all [tex]\( x \)[/tex] values less than [tex]\( -9 \)[/tex] and all [tex]\( x \)[/tex] values greater than [tex]\( 5 \)[/tex] satisfy the inequality [tex]\( |2x + 4| > 14 \)[/tex].
