To solve the absolute value inequality [tex]\( |x + 7| > 3 \)[/tex], we will follow these steps:
1. Understand the Absolute Value Inequality:
The expression [tex]\( |x + 7| > 3 \)[/tex] means that the distance of [tex]\( x + 7 \)[/tex] from 0 is greater than 3. This can be broken down into two possible cases:
- [tex]\( x + 7 > 3 \)[/tex]
- [tex]\( x + 7 < -3 \)[/tex]
2. Solve Each Case Separately:
- Case 1: [tex]\( x + 7 > 3 \)[/tex]
[tex]\[
x + 7 > 3
\][/tex]
Subtract 7 from both sides:
[tex]\[
x > -4
\][/tex]
- Case 2: [tex]\( x + 7 < -3 \)[/tex]
[tex]\[
x + 7 < -3
\][/tex]
Subtract 7 from both sides:
[tex]\[
x < -10
\][/tex]
3. Combine the Solutions:
We need [tex]\( x \)[/tex] to satisfy either of the conditions: [tex]\( x > -4 \)[/tex] or [tex]\( x < -10 \)[/tex]. Therefore, the solution is:
[tex]\[
x < -10 \quad \text{or} \quad x > -4
\][/tex]
4. Express the Solution in Interval Notation:
In interval notation, the solution is written as:
[tex]\[
(-\infty, -10) \cup (-4, \infty)
\][/tex]
So, the solution to the inequality [tex]\( |x + 7| > 3 \)[/tex] in interval notation is:
[tex]\[
(-\infty, -10) \cup (-4, \infty)
\][/tex]
