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The inequality is:

\[
x + 6 < 5x - 6
\]

### Step 1: Isolate the variable \( x \).

Subtract \( x \) from both sides:

\[
6 < 4x - 6
\]

### Step 2: Move constants to one side.

Add 6 to both sides:

\[
12 < 4x
\]

### Step 3: Solve for \( x \).

Divide both sides by 4:

\[
3 < x
\]

or equivalently:

\[
x > 3
\]

### Solution Set:
The solution set is all real numbers greater than 3, i.e., \( (3, \infty) \).

### Graphing the Solution:
On a number line:
- Draw an open circle at \( x = 3 \) (because \( x = 3 \) is not included in the solution).
- Shade the region to the right of 3, indicating that all numbers greater than 3 are part of the solution.

Would you like a graphical representation of the solution? If so, I can provide that as well.

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Do you have any questions? Here are some related questions to deepen your understanding:

1. How would the inequality change if it were \( x + 6 \leq 5x - 6 \)?
2. What happens to the solution set if you multiply both sides of an inequality by a negative number?
3. How do you interpret an open versus closed circle on a number line?
4. Can you provide another example of solving and graphing a linear inequality?
5. How can this concept be applied to inequalities involving quadratic expressions?

**Tip**: When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Solve and graph the solution set: x + 6 < 5x - 6

Learn how to solve and graph the linear inequality x + 6 < 5x - 6 with a step-by-step solution. This problem involves basic algebraic manipulation, isolating variables, and understanding inequality graphing. Suitable for students in Grades 9-10.

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