Sure, let's solve the inequality [tex]\( |x - 4| \leq 9 \)[/tex] step-by-step.
### Step 1: Understand Absolute Value Inequality
The absolute value inequality [tex]\( |x - 4| \leq 9 \)[/tex] means that the expression [tex]\( x - 4 \)[/tex] is within the distance of 9 units from 0. This can be translated into a compound inequality:
[tex]\[ -9 \leq x - 4 \leq 9 \][/tex]
### Step 2: Break Down the Compound Inequality
To solve the inequality [tex]\( -9 \leq x - 4 \leq 9 \)[/tex], we need to separate this into two simpler inequalities and solve each one:
1. [tex]\( x - 4 \leq 9 \)[/tex]
2. [tex]\( x - 4 \geq -9 \)[/tex]
### Step 3: Solve Each Inequality
Now let's solve each of these inequalities for [tex]\( x \)[/tex]:
#### Inequality 1: [tex]\( x - 4 \leq 9 \)[/tex]
To solve for [tex]\( x \)[/tex], add 4 to both sides of the inequality:
[tex]\[ x - 4 + 4 \leq 9 + 4 \][/tex]
[tex]\[ x \leq 13 \][/tex]
#### Inequality 2: [tex]\( x - 4 \geq -9 \)[/tex]
Similarly, add 4 to both sides of this inequality:
[tex]\[ x - 4 + 4 \geq -9 + 4 \][/tex]
[tex]\[ x \geq -5 \][/tex]
### Step 4: Combine the Results
Now that we have solved both inequalities, we can combine the results to find the solution to the original inequality:
[tex]\[ -5 \leq x \leq 13 \][/tex]
So, the solution to the inequality [tex]\( |x - 4| \leq 9 \)[/tex] is:
[tex]\[ \boxed{-5 \leq x \leq 13} \][/tex]
This means [tex]\( x \)[/tex] can be any value between [tex]\(-5\)[/tex] and [tex]\(13\)[/tex], inclusive.
