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Solve the absolute value inequality for [tex]x[/tex]:

[tex]\[ 2|3x + 9| \ \textless \ 36 \][/tex]

Select one:
a. [tex]\[ 9 \ \textless \ x \ \textless \ 36 \][/tex]
b. [tex]\[ -9 \ \textless \ x \ \textless \ 3 \][/tex]
c. [tex]\[ x \ \textless \ -9 \text{ or } x \ \textgreater \ 3 \][/tex]
d. [tex]\[ -18 \ \textless \ x \ \textless \ \frac{3}{4} \][/tex]

Sagot :

GinnyAnsweridn
To solve the absolute value inequality [tex]\(2|3x + 9| < 36\)[/tex], let's go through the process step by step.

1. Isolate the absolute value expression:
Divide both sides of the inequality by 2 to isolate the absolute value term.
[tex]\[ \frac{2|3x + 9|}{2} < \frac{36}{2} \][/tex]
Simplifying this gives:
[tex]\[ |3x + 9| < 18 \][/tex]

2. Rewrite the absolute value inequality as a compound inequality:
Recall that [tex]\(|A| < B\)[/tex] means [tex]\(-B < A < B\)[/tex]. Apply this principle to our problem:
[tex]\[ -18 < 3x + 9 < 18 \][/tex]

3. Solve the compound inequality:

First, we solve the left part of the inequality:
[tex]\[ -18 < 3x + 9 \][/tex]
Subtract 9 from both sides:
[tex]\[ -18 - 9 < 3x \][/tex]
Simplify:
[tex]\[ -27 < 3x \][/tex]
Divide by 3:
[tex]\[ -9 < x \][/tex]

Next, we solve the right part of the inequality:
[tex]\[ 3x + 9 < 18 \][/tex]
Subtract 9 from both sides:
[tex]\[ 3x < 18 - 9 \][/tex]
Simplify:
[tex]\[ 3x < 9 \][/tex]
Divide by 3:
[tex]\[ x < 3 \][/tex]

4. Combine the results:
We have two inequalities:
[tex]\[ -9 < x \][/tex]
and
[tex]\[ x < 3 \][/tex]

Combining these, we get the final solution as:
[tex]\[ -9 < x < 3 \][/tex]

Therefore, the correct answer is:
b. [tex]\(-9 < x < 3\)[/tex]