https://file.aiquickdraw.com/mathbot/file/1727998472114zbfh5vpy.jpg

The given inequality is \( |5 - 4x| > 8 \). Let's solve this step-by-step:

### Step 1: Break the absolute value inequality
For an inequality of the form \( |A| > B \), where \( B > 0 \), the solution is split into two cases:
1. \( A > B \)
2. \( A < -B \)

So, we break the inequality \( |5 - 4x| > 8 \) into two cases:
- \( 5 - 4x > 8 \)
- \( 5 - 4x < -8 \)

### Step 2: Solve each inequality

#### Case 1: \( 5 - 4x > 8 \)
Subtract 5 from both sides:
\[
-4x > 3
\]
Divide by \(-4\) (and reverse the inequality sign):
\[
x < -\frac{3}{4}
\]

#### Case 2: \( 5 - 4x < -8 \)
Subtract 5 from both sides:
\[
-4x < -13
\]
Divide by \(-4\) (and reverse the inequality sign):
\[
x > \frac{13}{4}
\]

### Step 3: Combine the solutions
The combined solution from both cases is:
\[
x < -\frac{3}{4} \quad \text{or} \quad x > \frac{13}{4}
\]

In interval notation, this is:
\[
(-\infty, -\frac{3}{4}) \cup (\frac{13}{4}, \infty)
\]

### Step 4: Match with the options
From the provided options in the image, the correct one is:
\[
(-\infty, -\frac{3}{4}) \cup (\frac{13}{4}, \infty)
\]

This corresponds to the second option in the list.

Would you like more details on any part of the solution? Here are five related questions to practice:

1. How do you solve absolute value inequalities of the form \( |A| < B \)?
2. What happens if \( B = 0 \) in an absolute value inequality \( |A| > B \)?
3. Can you graph the solution of \( |5 - 4x| > 8 \) on a number line?
4. How do you solve compound inequalities involving "and" or "or"?
5. What is the general rule for reversing inequality signs when multiplying or dividing by a negative number?

**Tip:** Always remember to reverse the inequality sign when dividing or multiplying both sides of an inequality by a negative number.

Solve the inequality |5 - 4x| > 8 and write the solution set in interval notation.

Discover how to solve the absolute value inequality |5 - 4x| > 8 step-by-step, and find the solution in interval notation. Learn key algebra concepts, including handling inequalities with absolute values and using interval notation.

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