Solve the following inequality. Write your answer in interval notation or if no solutions exist write no solutions.
3х + 4>1 or x-5≤3-3x

We are given the inequality:

\[
3x + 4 > 1 \quad \text{or} \quad x - 5 \leq 3 - 3x
\]

We will solve each part separately.

### 1. Solve \( 3x + 4 > 1 \):

Subtract 4 from both sides:
\[
3x > 1 - 4
\]
\[
3x > -3
\]
Divide by 3:
\[
x > -1
\]

Thus, the solution to this part is:
\[
x > -1
\]

### 2. Solve \( x - 5 \leq 3 - 3x \):

Add \( 3x \) to both sides:
\[
x + 3x - 5 \leq 3
\]
\[
4x - 5 \leq 3
\]
Add 5 to both sides:
\[
4x \leq 3 + 5
\]
\[
4x \leq 8
\]
Divide by 4:
\[
x \leq 2
\]

Thus, the solution to this part is:
\[
x \leq 2
\]

### Combine the results:

We now have two conditions:
\[
x > -1 \quad \text{or} \quad x \leq 2
\]
Since the union of these two conditions covers all real numbers (because \( x > -1 \) already includes all values greater than \(-1\) and the other inequality handles values less than or equal to 2), the solution is:

\[
\boxed{(-1, \infty)}
\]

### Conclusion:
The solution in interval notation is \( (-1, \infty) \).

Do you want further clarification on any steps, or have questions?  
Here are five related questions to expand your understanding:

1. How do you solve inequalities with "and" instead of "or"?
2. How do you graph solutions of inequalities on a number line?
3. Can you combine multiple inequalities into a compound inequality?
4. What is the difference between strict inequalities and non-strict inequalities?
5. How do you handle inequalities involving absolute values?

**Tip:** When solving compound inequalities, always pay attention to whether the conditions are connected by "and" or "or", as this changes the solution set.

Solve the following inequality. Write your answer in interval notation or if no solutions exist write no solutions. 3х + 4 > 1 or x - 5 ≤ 3 - 3x

Learn how to solve the compound inequality 3x + 4 > 1 or x - 5 ≤ 3 - 3x with step-by-step explanations. This problem involves solving linear inequalities and understanding the union of intervals. Ideal for students in Grades 9-11.

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