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What is the solution to [tex]\(9|x-8|\ \textless \ 36\)[/tex]?

A. [tex]\(4 \ \textless \ x \ \textless \ 12\)[/tex]
B. [tex]\(x \ \textless \ -4\)[/tex] or [tex]\(x \ \textgreater \ 12\)[/tex]
C. [tex]\(x \ \textgreater \ -12\)[/tex] or [tex]\(x \ \textless \ 8\)[/tex]
D. [tex]\(-4 \ \textless \ x \ \textless \ 8\)[/tex]

Sagot :

GinnyAnswer
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]
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