Solve the inequality. (Enter your answer using interval notation.)
|5x − 1| < 7

We are given the inequality:

\[
|5x - 1| < 7
\]

To solve this, we must consider the definition of absolute value. The inequality \( |A| < B \) (where \( A \) is an expression and \( B \) is a positive number) is equivalent to the double inequality:

\[
-B < A < B
\]

### Step 1: Rewrite the inequality without the absolute value

The inequality \( |5x - 1| < 7 \) becomes:

\[
-7 < 5x - 1 < 7
\]

### Step 2: Solve for \( x \)

First, isolate \( 5x \) by adding 1 to all parts of the inequality:

\[
-7 + 1 < 5x < 7 + 1
\]

\[
-6 < 5x < 8
\]

Now, divide the entire inequality by 5 to isolate \( x \):

\[
\frac{-6}{5} < x < \frac{8}{5}
\]

This simplifies to:

\[
-\frac{6}{5} < x < \frac{8}{5}
\]

### Step 3: Write the solution in interval notation

The solution is the set of \( x \)-values between \( -\frac{6}{5} \) and \( \frac{8}{5} \), not including the endpoints. So, the solution in interval notation is:

\[
\left( -\frac{6}{5}, \frac{8}{5} \right)
\]

Let me know if you'd like further details or explanations!

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Here are 5 related questions for you to consider:

1. How do you solve inequalities involving \( \geq \) or \( \leq \)?
2. How does the solution change if the inequality is \( |5x - 1| \leq 7 \)?
3. How would you solve \( |5x - 1| > 7 \)?
4. What is the general method for solving inequalities with absolute values?
5. How do you graph the solution of \( |5x - 1| < 7 \) on a number line?

**Tip:** Always check your solution by plugging in values from within and outside your interval to verify if the inequality holds true.

Solve the inequality. (Enter your answer using interval notation.) |5x − 1| < 7

Learn how to solve the absolute value inequality |5x − 1| < 7 using interval notation with step-by-step guidance. This problem covers important algebra concepts, including solving absolute value and linear inequalities. Suitable for students in Grades 8-10.

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