Solve: 3𝑦 + 𝑥 < 7𝑥 + 6.

Responses

y < 2𝑥 + 2

y < 2𝑥 + 2

y < 2𝑥 + 1

y < 2𝑥 + 1

y < 3𝑥 + 1

y < 3𝑥 + 1

y < −2𝑥 + 2

Let's solve the inequality step by step:

### Given inequality: 
\[ 3y + x < 7x + 6 \]

### Step 1: Isolate terms with \(y\)
We want to move the \(x\)-related terms to the other side of the inequality.

\[
3y < 7x - x + 6
\]

Simplifying the terms:
\[
3y < 6x + 6
\]

### Step 2: Solve for \(y\)
Now, divide both sides by 3 to isolate \(y\):

\[
y < \frac{6x + 6}{3}
\]

Simplifying the right-hand side:

\[
y < 2x + 2
\]

### Conclusion:
The solution is:
\[
y < 2x + 2
\]

This corresponds to the first two responses.

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Would you like a more detailed explanation on this? Here are five additional questions based on this topic:

1. How do we solve an inequality differently from an equation?
2. What changes when we multiply or divide an inequality by a negative number?
3. How would this inequality look if solved for \(x\) instead of \(y\)?
4. Can inequalities like this be represented graphically? If so, how?
5. How do we deal with compound inequalities involving both \(x\) and \(y\)?

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

Learn how to solve the linear inequality 3y + x < 7x + 6 with a step-by-step solution. This problem focuses on algebraic concepts such as solving inequalities and simplifying linear equations. Suitable for students in Grades 8-10.

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