Certainly! Let's solve the given absolute value inequality step-by-step.
Given Inequality:
[tex]\[ 4|x-9| \geq 16 \][/tex]
Step 1: Isolate the absolute value expression.
Divide both sides of the inequality by 4:
[tex]\[ |x-9| \geq \frac{16}{4} \][/tex]
[tex]\[ |x-9| \geq 4 \][/tex]
Step 2: Consider the definition of absolute value.
For any number [tex]\( y \)[/tex], [tex]\( |y| \geq a \)[/tex] implies two conditions:
[tex]\[ y \geq a \quad \text{or} \quad y \leq -a \][/tex]
In our case, [tex]\( y = x - 9 \)[/tex] and [tex]\( a = 4 \)[/tex]. Therefore, we have:
[tex]\[ x - 9 \geq 4 \quad \text{or} \quad x - 9 \leq -4 \][/tex]
Step 3: Solve each condition separately.
1. Solve [tex]\( x - 9 \geq 4 \)[/tex]:
[tex]\[ x - 9 \geq 4 \][/tex]
Add 9 to both sides:
[tex]\[ x \geq 4 + 9 \][/tex]
[tex]\[ x \geq 13 \][/tex]
2. Solve [tex]\( x - 9 \leq -4 \)[/tex]:
[tex]\[ x - 9 \leq -4 \][/tex]
Add 9 to both sides:
[tex]\[ x \leq -4 + 9 \][/tex]
[tex]\[ x \leq 5 \][/tex]
Solution:
The solution to the inequality [tex]\( 4|x-9| \geq 16 \)[/tex] is:
[tex]\[ x \geq 13 \quad \text{or} \quad x \leq 5 \][/tex]
