To solve the inequality [tex]\(|6x - 2| - 6 < 10\)[/tex], we can follow these steps:
1. Simplify the Inequality:
We start with the inequality:
[tex]\[
|6x - 2| - 6 < 10
\][/tex]
Add 6 to both sides to isolate the absolute value expression:
[tex]\[
|6x - 2| < 16
\][/tex]
2. Interpret the Absolute Value Inequality:
The inequality [tex]\(|6x - 2| < 16\)[/tex] means that the expression inside the absolute value lies within the range [tex]\(-16\)[/tex] to [tex]\(16\)[/tex]. Therefore, we can rewrite this as:
[tex]\[
-16 < 6x - 2 < 16
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
We now handle the combined inequality by splitting it into two separate inequalities and solving them:
- First, solve [tex]\(-16 < 6x - 2\)[/tex]:
[tex]\[
-16 < 6x - 2
\][/tex]
Add 2 to both sides:
[tex]\[
-14 < 6x
\][/tex]
Divide by 6:
[tex]\[
-\frac{14}{6} < x
\][/tex]
Simplify the fraction:
[tex]\[
-\frac{7}{3} < x
\][/tex]
- Next, solve [tex]\(6x - 2 < 16\)[/tex]:
[tex]\[
6x - 2 < 16
\][/tex]
Add 2 to both sides:
[tex]\[
6x < 18
\][/tex]
Divide by 6:
[tex]\[
x < 3
\][/tex]
4. Combine the Solutions:
Combining the results from both parts, we get:
[tex]\[
-\frac{7}{3} < x < 3
\][/tex]
Hence, the solution to the inequality [tex]\(|6x - 2| - 6 < 10\)[/tex] is:
[tex]\[
-\frac{7}{3} < x < 3
\][/tex]
