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.

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.