Certainly! Let's solve the inequality [tex]\( |3x| + 4 < 10 \)[/tex] step by step.
1. Start with the given inequality:
[tex]\[
|3x| + 4 < 10
\][/tex]
2. Isolate the absolute value term:
[tex]\[
|3x| + 4 - 4 < 10 - 4
\][/tex]
Simplifying, we get:
[tex]\[
|3x| < 6
\][/tex]
3. Understand the meaning of the absolute value inequality:
The inequality [tex]\( |3x| < 6 \)[/tex] means that the expression inside the absolute value [tex]\(3x\)[/tex] lies between [tex]\(-6\)[/tex] and [tex]\(6\)[/tex].
4. Write the compound inequality:
[tex]\[
-6 < 3x < 6
\][/tex]
5. Solve for [tex]\(x\)[/tex] by dividing every part of the inequality by 3:
[tex]\[
\frac{-6}{3} < \frac{3x}{3} < \frac{6}{3}
\][/tex]
Simplifying each term, we get:
[tex]\[
-2 < x < 2
\][/tex]
Thus, the solution to the inequality [tex]\( |3x| + 4 < 10 \)[/tex] is:
[tex]\[
-2 < x < 2
\][/tex]
This means that [tex]\(x\)[/tex] must lie within the interval [tex]\((-2, 2)\)[/tex].
