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The problem asks you to solve the inequality \( x + 1 > 2x - 6 \) graphically by plotting the lines \( y_1 = x + 1 \) and \( y_2 = 2x - 6 \), and then determining the solution set. Here's a step-by-step breakdown:

### Step 1: Set up the inequality
You're solving the inequality \( x + 1 > 2x - 6 \).

This can be rewritten as:

\[
x + 1 > 2x - 6
\]

### Step 2: Simplify the inequality
Move all terms involving \( x \) to one side:

\[
1 + 6 > 2x - x
\]

\[
7 > x \quad \text{or} \quad x < 7
\]

### Step 3: Graph the equations
You are given the two equations:
- \( y_1 = x + 1 \)
- \( y_2 = 2x - 6 \)

These are two straight lines, and you need to find where \( y_1 \) is greater than \( y_2 \). 

### Step 4: Find the intersection point
The intersection point of the two lines occurs when \( x + 1 = 2x - 6 \). Solving for \( x \):

\[
x + 1 = 2x - 6
\]

\[
1 + 6 = 2x - x
\]

\[
x = 7
\]

The two lines intersect at \( x = 7 \).

### Step 5: Determine the solution set
From the inequality, you want to find when \( y_1 \) is greater than \( y_2 \), which occurs when \( x < 7 \). So, the solution set in interval notation is:

\[
(-\infty, 7)
\]

### Conclusion:
The solution set is \( (-\infty, 7) \).

Would you like more details or have any questions about this solution?

Here are 5 related questions:
1. How do you graph inequalities on a coordinate plane?
2. What is the meaning of intersection points in solving inequalities graphically?
3. How do you solve compound inequalities?
4. How does solving inequalities differ from solving equations?
5. How do you express solutions to inequalities using interval notation?

**Tip**: When graphing inequalities, always check the behavior of the graph above and below the intersection point to find the correct regions where the inequality holds.

Solve graphically: x + 1 > 2x - 6. Graph y1 = x + 1 and y2 = 2x - 6 using the graphing tool. What is the solution set?

Learn how to graph and solve the inequality x + 1 > 2x - 6. This problem involves graphing linear equations and determining the solution set by analyzing the intersection of the two lines y = x + 1 and y = 2x - 6. Suitable for students in Grades 7-9.

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