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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]\[ -18 \ \textless \ x \ \textless \ \frac{3}{4} \][/tex]
c. [tex]\[ x \ \textless \ -9 \text{ or } x \ \textgreater \ 3 \][/tex]
d. [tex]\[ -9 \ \textless \ x \ \textless \ 3 \][/tex]

Sagot :

GinnyAnswer
To solve the absolute value inequality [tex]\(2|3x + 9| < 36\)[/tex], follow these steps:

1. Isolate the absolute value expression:
[tex]\[ 2|3x + 9| < 36 \][/tex]

2. Divide both sides by 2 to simplify:
[tex]\[ |3x + 9| < 18 \][/tex]

3. Understand the meaning of the absolute value inequality:
[tex]\[ |3x + 9| < 18 \][/tex]
This implies that the expression inside the absolute value, [tex]\(3x + 9\)[/tex], must lie between -18 and 18. Hence, we can write:
[tex]\[ -18 < 3x + 9 < 18 \][/tex]

4. Solve the compound inequality:
- First, subtract 9 from all parts of the inequality:
[tex]\[ -18 - 9 < 3x < 18 - 9 \][/tex]
Simplifies to:
[tex]\[ -27 < 3x < 9 \][/tex]

- Then, divide all parts by 3:
[tex]\[ \frac{-27}{3} < x < \frac{9}{3} \][/tex]
Simplifies to:
[tex]\[ -9 < x < 3 \][/tex]

Therefore, the solution to the inequality [tex]\(2|3x + 9| < 36\)[/tex] is:
[tex]\[ -9 < x < 3 \][/tex]

The correct answer is:
d. [tex]\(-9 < x < 3\)[/tex]
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