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What is the solution to [tex]|x-9|-3\ \textless \ 1[/tex]?

A. [tex]x\ \textless \ 5[/tex] or [tex]x\ \textgreater \ 13[/tex]

B. [tex]5\ \textless \ x\ \textless \ 13[/tex]

C. [tex]-13\ \textless \ x\ \textless \ -5[/tex]

D. [tex]x \leq -13[/tex] or [tex]x \geq -5[/tex]



Answer :

GinnyAnswer
To solve the inequality [tex]\( |x - 9| - 3 < 1 \)[/tex], let's go through it step-by-step:

1. Rewrite the Inequality:
The given inequality is [tex]\( |x - 9| - 3 < 1 \)[/tex]. First, isolate the absolute value term:
[tex]\[ |x - 9| - 3 < 1 \][/tex]
Add 3 to both sides:
[tex]\[ |x - 9| < 4 \][/tex]

2. Definition of Absolute Value:
Recall that for any real number [tex]\( a \)[/tex], [tex]\( |a| < b \)[/tex] implies:
[tex]\[ -b < a < b \][/tex]
Apply this rule to our inequality [tex]\( |x - 9| < 4 \)[/tex]:
[tex]\[ -4 < x - 9 < 4 \][/tex]

3. Solving the Compound Inequality:
We now solve the compound inequality for [tex]\( x \)[/tex]:
[tex]\[ -4 < x - 9 < 4 \][/tex]
Add 9 to all parts of the inequality:
[tex]\[ -4 + 9 < x - 9 + 9 < 4 + 9 \][/tex]
Simplify:
[tex]\[ 5 < x < 13 \][/tex]

Therefore, the solution to the inequality [tex]\( |x - 9| - 3 < 1 \)[/tex] is:
[tex]\[ 5 < x < 13 \][/tex]

The correct answer is:
B. [tex]\( 5 < x < 13 \)[/tex]

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