2x-6 is less than or equal to -10 or 3x + 1 >16

To solve the compound inequality \(2x - 6 \leq -10 \, \text{or} \, 3x + 1 > 16\), we will solve each inequality separately and then combine the results.

### 1. Solving \(2x - 6 \leq -10\):
\[
2x - 6 \leq -10
\]
Add 6 to both sides:
\[
2x \leq -4
\]
Divide by 2:
\[
x \leq -2
\]

### 2. Solving \(3x + 1 > 16\):
\[
3x + 1 > 16
\]
Subtract 1 from both sides:
\[
3x > 15
\]
Divide by 3:
\[
x > 5
\]

### Final solution:
The compound inequality uses the word "or," which means we combine both solution sets:
- From \(2x - 6 \leq -10\), we get \(x \leq -2\).
- From \(3x + 1 > 16\), we get \(x > 5\).

Thus, the solution is \(x \leq -2\) or \(x > 5\).

Would you like further clarification or details on any part of this process?

Here are some related questions for practice:
1. How would the solution change if the inequality was "and" instead of "or"?
2. Solve \(4x - 7 \leq -3 \, \text{or} \, 2x + 5 > 11\).
3. What happens if both inequalities have the same solution range?
4. How do you graph \(x \leq -2\) or \(x > 5\) on a number line?
5. Solve the compound inequality \(5x - 4 \geq 6 \, \text{and} \, x + 2 < 10\).

**Tip**: Always pay close attention to the conjunction "and" or "or" in compound inequalities, as they dictate how solution sets are combined.

2x - 6 is less than or equal to -10 or 3x + 1 > 16

This guide explains how to solve the compound inequality 2x - 6 ≤ -10 or 3x + 1 > 16. Learn step-by-step how to handle each inequality separately and then combine the solution sets. Suitable for students in Grades 8-10, this tutorial covers key algebraic concepts, including inequalities and compound inequalities.

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