Question

Which graph shows the solution of −6(n−1)<3and2(n+1)>0?

Answer 1

To solve the inequalities \( -6(n-1) < 3 \) and \( 2(n+1) > 0 \), we will solve each inequality step by step.

### First Inequality: \( -6(n-1) < 3 \)

1. Distribute the \(-6\):
\[
-6n + 6 < 3
\]

2. Subtract 6 from both sides:
\[
-6n < 3 - 6
\]
\[
-6n < -3
\]

3. Divide by \(-6\) and remember to reverse the inequality sign:
\[
n > \frac{-3}{-6}
\]
\[
n > \frac{1}{2}
\]

### Second Inequality: \( 2(n+1) > 0 \)

1. Distribute the \(2\):
\[
2n + 2 > 0
\]

2. Subtract \(2\) from both sides:
\[
2n > -2
\]

3. Divide by \(2\):
\[
n > -1
\]

### Combined Solution

Now we combine the results of both inequalities:
1. \( n > \frac{1}{2} \)
2. \( n > -1 \)

The more restrictive solution is \( n > \frac{1}{2} \).

### Graph Interpretation

On a number line, this solution \( n > \frac{1}{2} \) will be represented as:
- An open circle at \( \frac{1}{2} \) (since \( n > \frac{1}{2} \) does not include \( \frac{1}{2} \)),
- A shaded region extending to the right (indicating all numbers greater than \( \frac{1}{2} \)).

If you have specific graphs to choose from, look for the one that has these characteristics.