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.