To solve the inequality [tex]\(|4x - 3| < 2\)[/tex], we need to consider the definition and properties of absolute values. The absolute value inequality [tex]\(|A| < B\)[/tex] implies [tex]\(-B < A < B\)[/tex]. Therefore, we will break down the inequality step-by-step:
1. Rewrite the inequality:
[tex]\[
|4x - 3| < 2
\][/tex]
This means:
[tex]\[
-2 < 4x - 3 < 2
\][/tex]
2. Solve the compound inequality:
To solve the inequality [tex]\(-2 < 4x - 3 < 2\)[/tex], we'll split it into two separate inequalities and solve each:
[tex]\[
-2 < 4x - 3
\][/tex]
[tex]\[
4x - 3 < 2
\][/tex]
3. Solve [tex]\(-2 < 4x - 3\)[/tex]:
[tex]\[
-2 < 4x - 3
\][/tex]
Add 3 to both sides:
[tex]\[
-2 + 3 < 4x
\][/tex]
Simplify:
[tex]\[
1 < 4x
\][/tex]
Divide by 4:
[tex]\[
\frac{1}{4} < x
\][/tex]
Which can also be written as:
[tex]\[
x > \frac{1}{4}
\][/tex]
4. Solve [tex]\(4x - 3 < 2\)[/tex]:
[tex]\[
4x - 3 < 2
\][/tex]
Add 3 to both sides:
[tex]\[
4x < 2 + 3
\][/tex]
Simplify:
[tex]\[
4x < 5
\][/tex]
Divide by 4:
[tex]\[
x < \frac{5}{4}
\][/tex]
5. Combine the inequalities:
Combine the results from both solved inequalities:
[tex]\[
\frac{1}{4} < x < \frac{5}{4}
\][/tex]
6. Write the solution in interval notation:
The solution to the inequality [tex]\(|4x - 3| < 2\)[/tex] is expressed in interval notation as:
[tex]\[
\left( \frac{1}{4}, \frac{5}{4} \right)
\][/tex]
Thus, the solution in interval notation is [tex]\(\left( \frac{1}{4}, \frac{5}{4} \right)\)[/tex].
