To solve the inequality \( |4x - 7| > 3 \), we need to consider the definition of absolute value and solve two separate inequalities:

### Step 1: Set up the inequalities
The inequality \( |4x - 7| > 3 \) can be split into two cases:

1. \( 4x - 7 > 3 \)
2. \( 4x - 7 < -3 \)

### Step 2: Solve each inequality separately

**Case 1: \( 4x - 7 > 3 \)**

\[
4x > 3 + 7
\]
\[
4x > 10
\]
\[
x > \frac{10}{4}
\]
\[
x > 2.5
\]

**Case 2: \( 4x - 7 < -3 \)**

\[
4x < -3 + 7
\]
\[
4x < 4
\]
\[
x < \frac{4}{4}
\]
\[
x < 1
\]

### Step 3: Combine the solutions
The solution to the inequality is the union of the solutions from the two cases. So, the final answer is:

\[
x < 1 \quad \text{or} \quad x > 2.5
\]

### Solution in Interval Notation:
\[
(-\infty, 1) \cup (2.5, \infty)
\]

Would you like more details on this, or do you have any questions?

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Here are 5 related questions you might find interesting:

1. How would you solve the inequality \( |3x + 2| \leq 5 \)?
2. What is the solution to \( |2x - 1| = 7 \)?
3. Can you solve \( |x^2 - 4| > 5 \)?
4. How would you graph the solution to \( |5x - 3| \leq 2 \)?
5. What is the difference between solving \( |x + 1| > 4 \) and \( |x + 1| \geq 4 \)?

**Tip:** When solving absolute value inequalities, always remember to consider both the positive and negative scenarios by splitting the inequality into two cases.

Learn how to solve the absolute value inequality |4x − 7| > 3 step-by-step. This problem involves understanding absolute value, inequalities, and interval notation. Suitable for high school students (Grades 9-12).

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