We are tasked with solving the following compound inequality and expressing the solution in both interval and graph form:

$x + 1 > 5 \quad \text{or} \quad -3x + 2 \geq 8$

Step 1: Solve each inequality separately.

1. First Inequality:
   $x + 1 > 5$ Subtract 1 from both sides: $x > 4$

2. Second Inequality: $-3x + 2 \geq 8$ Subtract 2 from both sides: $-3x \geq 6$ Divide by $-3$ (note that the inequality sign flips when dividing by a negative number): $x \leq -2$

Step 2: Combine the solutions.

The combined inequality is: $x > 4 \quad \text{or} \quad x \leq -2$

Step 3: Write the solution in interval form.

• The solution for $x > 4$ is $(4, \infty)$.
• The solution for $x \leq -2$ is $(-\infty, -2]$.

Thus, the solution in interval form is: $(-\infty, -2] \cup (4, \infty)$

Step 4: Graph the solution.

The graph will consist of two parts:

• A closed dot at $x = -2$ and a shaded line extending to the left.
• An open dot at $x = 4$ and a shaded line extending to the right.

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Would you like me to provide a graph of this solution as well?

Related questions:

1. How do you solve compound inequalities with "and" instead of "or"?
2. What happens if both inequalities overlap?
3. How does solving inequalities differ from solving regular equations?
4. Can compound inequalities have more than two parts?
5. What is the difference between strict inequalities ($>$) and non-strict inequalities ($\geq$)?

Tip:

Always remember to flip the inequality sign when dividing or multiplying by a negative number!