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The problem asks to solve the inequality \( |4x - 2| + 1 > 6 \) by graphing. Let me guide you through the steps to solve it graphically.

### Step 1: Simplify the inequality
Start by isolating the absolute value:
\[
|4x - 2| + 1 > 6
\]
Subtract 1 from both sides:
\[
|4x - 2| > 5
\]

### Step 2: Set up the conditions for the absolute value
The expression \( |4x - 2| > 5 \) means two possibilities:
1. \( 4x - 2 > 5 \)
2. \( 4x - 2 < -5 \)

### Step 3: Solve each case

#### Case 1:
\[
4x - 2 > 5
\]
Add 2 to both sides:
\[
4x > 7
\]
Divide by 4:
\[
x > \frac{7}{4} \quad \text{or} \quad x > 1.75
\]

#### Case 2:
\[
4x - 2 < -5
\]
Add 2 to both sides:
\[
4x < -3
\]
Divide by 4:
\[
x < -\frac{3}{4} \quad \text{or} \quad x < -0.75
\]

### Step 4: Graph the solution
The solution is the union of two intervals:
\[
x < -0.75 \quad \text{or} \quad x > 1.75
\]
To graph this:
- Mark \( x = -0.75 \) and \( x = 1.75 \) on the number line.
- Shade the region to the left of \( x = -0.75 \) and the region to the right of \( x = 1.75 \).
- The points at \( x = -0.75 \) and \( x = 1.75 \) are **open circles** since the inequality is strict (i.e., no equality).

Would you like more details on how to plot this graph, or do you have any questions about the solution?

### Related Questions:
1. How do you solve absolute value inequalities algebraically?
2. Can you graph inequalities involving absolute values on a number line?
3. What is the difference between strict inequalities and non-strict inequalities?
4. How do you check the solution to an inequality by substitution?
5. What happens if the inequality involves a negative absolute value?

### Tip:
When solving absolute value inequalities, remember that they split into two separate cases, one for the positive and one for the negative.

Solve |4x - 2| + 1 > 6 by graphing. Round to the nearest hundredth if necessary.

Learn how to solve the inequality |4x - 2| + 1 > 6 by graphing and find the solution in two intervals. This problem covers key algebraic concepts related to absolute value inequalities and is suitable for students in Grades 9-11.

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