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

[tex]|3x - 1| \ \textgreater \ 11[/tex]

Select one:
a. [tex]-4 \ \textless \ x \ \textless \ 4[/tex]
b. [tex]x \ \textgreater \ 4[/tex] or [tex]x \ \textless \ -\frac{10}{3}[/tex]
c. [tex]10 \ \textless \ x \ \textless \ -\frac{10}{3}[/tex]
d. [tex]-\frac{10}{3} \ \textless \ x \ \textless \ 4[/tex]

Sagot :

GinnyAnsweridn
To solve the absolute value inequality [tex]\( |3x - 1| > 11 \)[/tex], we can break it down into two separate inequalities due to the nature of the absolute value function.

The absolute value inequality [tex]\( |A| > B \)[/tex] is equivalent to the two inequalities:
[tex]\[ A > B \text{ or } A < -B \][/tex]

Applying this to our inequality [tex]\( |3x - 1| > 11 \)[/tex], we get:
[tex]\[ 3x - 1 > 11 \text{ or } 3x - 1 < -11 \][/tex]

Let's solve each inequality separately:

1. [tex]\( 3x - 1 > 11 \)[/tex]

Add 1 to both sides:
[tex]\[ 3x > 12 \][/tex]

Divide both sides by 3:
[tex]\[ x > 4 \][/tex]

2. [tex]\( 3x - 1 < -11 \)[/tex]

Add 1 to both sides:
[tex]\[ 3x < -10 \][/tex]

Divide both sides by 3:
[tex]\[ x < -\frac{10}{3} \][/tex]

Thus, the solutions to the inequalities [tex]\( |3x - 1| > 11 \)[/tex] are:
[tex]\[ x > 4 \text{ or } x < -\frac{10}{3} \][/tex]

So the correct solution for the absolute value inequality is:
[tex]\[ x > 4 \text{ or } x < -\frac{10}{3} \][/tex]

This corresponds to option (b):
[tex]\[ \text{b. } x > 4 \text{ or } x < -\frac{10}{3} \][/tex]
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