Sure, let's solve the inequality [tex]\( |x+8| > 3 \)[/tex] step by step.
### Step 1: Understand the Absolute Value Inequality
The absolute value inequality [tex]\( |x+8| > 3 \)[/tex] means we have two separate inequalities to consider:
- [tex]\( x + 8 > 3 \)[/tex]
- [tex]\( x + 8 < -3 \)[/tex]
### Step 2: Solve Each Inequality Separately
#### Inequality 1: [tex]\( x + 8 > 3 \)[/tex]
1. Subtract 8 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x + 8 > 3 \implies x > 3 - 8
\][/tex]
2. Simplify the right side:
[tex]\[
x > -5
\][/tex]
#### Inequality 2: [tex]\( x + 8 < -3 \)[/tex]
1. Subtract 8 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x + 8 < -3 \implies x < -3 - 8
\][/tex]
2. Simplify the right side:
[tex]\[
x < -11
\][/tex]
### Step 3: Combine the Solutions
The solution to the absolute value inequality [tex]\( |x + 8| > 3 \)[/tex] includes both of the above results:
[tex]\[
x > -5 \quad \text{or} \quad x < -11
\][/tex]
### Step 4: Write the Solution in Interval Notation
1. The inequality [tex]\( x > -5 \)[/tex] corresponds to the interval [tex]\( (-5, \infty) \)[/tex].
2. The inequality [tex]\( x < -11 \)[/tex] corresponds to the interval [tex]\( (-\infty, -11) \)[/tex].
### Step 5: Combine Intervals
Put together, the solution to the inequality [tex]\( |x + 8| > 3 \)[/tex] is the union of these two intervals:
[tex]\[
(-\infty, -11) \cup (-5, \infty)
\][/tex]
Therefore, the solution to the inequality [tex]\( |x+8| > 3 \)[/tex] is the set of all [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is in the interval [tex]\( (-\infty, -11) \)[/tex] or [tex]\( x \)[/tex] is in the interval [tex]\( (-5, \infty) \)[/tex].
