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Solve for the set of solutions for:

[tex]\[ \left|x + \frac{1}{4} \right| \ \textgreater \ \frac{7}{4} \][/tex]

A. [tex]\( x \in (-\infty, -2) \cup \left( \frac{3}{2}, \infty \right) \)[/tex]

B. [tex]\( x \in (-\infty, -2) \cup \left( \frac{1}{2}, \infty \right) \)[/tex]

C. [tex]\( x \in (-\infty, -1) \cup \left( \frac{3}{2}, \infty \right) \)[/tex]

D. [tex]\( x \in (-\infty, -4) \cup \left( \frac{1}{2}, \infty \right) \)[/tex]



Answer :

GinnyAnswer
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).

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