We are given two inequalities:

\[
- x + 6 > 2 \quad \text{or} \quad - x + 6 \leq -5
\]

Let's solve each inequality separately.

### Inequality 1: \( -x + 6 > 2 \)

1. Subtract 6 from both sides:
   \[
   -x > 2 - 6
   \]
   \[
   -x > -4
   \]

2. Divide both sides by \(-1\) (remember to reverse the inequality when dividing by a negative number):
   \[
   x < 4
   \]

### Inequality 2: \( -x + 6 \leq -5 \)

1. Subtract 6 from both sides:
   \[
   -x \leq -5 - 6
   \]
   \[
   -x \leq -11
   \]

2. Divide both sides by \(-1\) (again, reverse the inequality):
   \[
   x \geq 11
   \]

### Final Solution

The solution to the compound inequality is:

\[
x < 4 \quad \text{or} \quad x \geq 11
\]

This means \( x \) can be any number less than 4 or any number greater than or equal to 11.

Would you like more details or have any questions? Here are 5 related questions:

1. How do you handle compound inequalities that involve "and" instead of "or"?
2. What happens when you multiply or divide both sides of an inequality by a negative number?
3. Can you graph the solution of this inequality on a number line?
4. How do you solve an inequality if the variable appears on both sides of the inequality?
5. How do you check if a particular number is a solution to an inequality?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Learn how to solve the compound inequalities −x + 6 > 2 or −x + 6 ≤ −5 with a detailed, step-by-step explanation. Understand how to handle inequalities involving negative coefficients and apply key algebraic rules. Suitable for students in Grades 9-10.

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