To solve the inequality [tex]\( 9|x-8| < 36 \)[/tex], follow these step-by-step instructions:
1. Divide both sides by 9:
[tex]\( \frac{9|x-8|}{9} < \frac{36}{9} \)[/tex]
Simplifies to:
[tex]\( |x-8| < 4 \)[/tex]
2. Interpret the absolute value inequality:
The inequality [tex]\( |x-8| < 4 \)[/tex] means that the distance between [tex]\( x \)[/tex] and 8 is less than 4. This can be translated into a compound inequality:
[tex]\[ -4 < x-8 < 4 \][/tex]
3. Isolate [tex]\( x \)[/tex] by adding 8 to all parts of the inequality:
[tex]\[ -4 + 8 < x - 8 + 8 < 4 + 8 \][/tex]
Simplifies to:
[tex]\[ 4 < x < 12 \][/tex]
Therefore, the solution to the inequality [tex]\( 9|x-8| < 36 \)[/tex] is:
[tex]\[ 4 < x < 12 \][/tex]
From the given options, the correct answer is:
[tex]\[ 4 < x < 12 \][/tex]
