Solve the inequality. Check all answers that are correct.

4(3x-5)<-31+12x

Let's solve the inequality step by step:

We are given:
\[
4(3x - 5) < -31 + 12x
\]

### Step 1: Distribute the 4 on the left side.
\[
4 \cdot 3x - 4 \cdot 5 < -31 + 12x
\]
\[
12x - 20 < -31 + 12x
\]

### Step 2: Subtract \(12x\) from both sides to eliminate it on the left side.
\[
12x - 20 - 12x < -31 + 12x - 12x
\]
\[
-20 < -31
\]

### Step 3: Analyze the result.
We are left with:
\[
-20 < -31
\]
This is a false statement, which means there are no solutions for \(x\) that satisfy the inequality.

### Conclusion:
The inequality has no solutions.

Would you like more details on the process? 

Here are 5 related questions:
1. What happens if an inequality simplifies to a false statement?
2. How do we approach solving linear inequalities with variables on both sides?
3. How can we verify inequality solutions?
4. What is the difference between an equation and an inequality in algebra?
5. How would the solution change if the inequality sign was \( \leq \) instead of \( < \)?

**Tip:** When solving inequalities, always check if the inequality simplifies to a true or false statement; this determines whether there are no solutions or all real numbers are solutions!

Learn how to solve the linear inequality 4(3x-5)<-31+12x step-by-step. This problem covers key algebraic concepts, including the distributive property and the properties of inequalities. Suitable for students in Grades 8-10.

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