To find the solution to the inequality [tex]\(\left|x+\frac{1}{4}\right|>\frac{7}{4}\)[/tex], we'll follow these steps:
1. Understand the Structure of Absolute Value Inequality:
The inequality [tex]\(\left|x+\frac{1}{4}\right|>\frac{7}{4}\)[/tex] implies that [tex]\(x+\frac{1}{4}\)[/tex] can be greater than [tex]\(\frac{7}{4}\)[/tex] or less than [tex]\(-\frac{7}{4}\)[/tex]. This gives us two separate inequalities to solve:
[tex]\[
x + \frac{1}{4} > \frac{7}{4} \quad \text{or} \quad x + \frac{1}{4} < -\frac{7}{4}
\][/tex]
2. Solve the First Inequality:
[tex]\[
x + \frac{1}{4} > \frac{7}{4}
\][/tex]
Subtract [tex]\(\frac{1}{4}\)[/tex] from both sides:
[tex]\[
x > \frac{7}{4} - \frac{1}{4}
\][/tex]
Simplify the right-hand side:
[tex]\[
x > \frac{6}{4} = \frac{3}{2}
\][/tex]
3. Solve the Second Inequality:
[tex]\[
x + \frac{1}{4} < -\frac{7}{4}
\][/tex]
Subtract [tex]\(\frac{1}{4}\)[/tex] from both sides:
[tex]\[
x < -\frac{7}{4} - \frac{1}{4}
\][/tex]
Simplify the right-hand side:
[tex]\[
x < -\frac{8}{4} = -2
\][/tex]
4. Combine the Results:
The solutions to the inequalities are:
[tex]\[
x > \frac{3}{2} \quad \text{or} \quad x < -2
\][/tex]
In interval notation, this is:
[tex]\[
x \in (-\infty, -2) \cup \left( \frac{3}{2}, \infty \right)
\][/tex]
5. Match with the Given Options:
Comparing this to the provided choices, we find that the correct answer is:
[tex]\[
\boxed{x \in(-\infty, -2) \cup \left(\frac{3}{2}, \infty \right)}
\][/tex]
Therefore, the correct option is (A).
