Which is the graph of the linear inequality 2x – 3y < 12?

On a coordinate plane, a solid straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the right of the line is shaded. 

On a coordinate plane, a dashed straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the right of the line is shaded.

On a coordinate plane, a dashed straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the left of the line is shaded.

On a coordinate plane, a solid straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the left of the line is shaded.


To determine the correct graph for the inequality \(2x - 3y < 12\), we can first rewrite the inequality in slope-intercept form, i.e., \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

### Step 1: Rewrite the inequality
Start by solving for \(y\):
\[
2x - 3y < 12
\]
Subtract \(2x\) from both sides:
\[
-3y < -2x + 12
\]
Now, divide by \(-3\) (remember that dividing by a negative number flips the inequality sign):
\[
y > \frac{2}{3}x - 4
\]
So, the inequality becomes:
\[
y > \frac{2}{3}x - 4
\]

### Step 2: Analyze the components
1. **Slope and intercept**: The slope is \(\frac{2}{3}\) (positive slope), and the y-intercept is \(-4\) (the line passes through the point \((0, -4)\)).
2. **Dashed line**: Since the inequality is a strict inequality (\(>\), not \(\geq\)), the line is dashed, not solid.
3. **Shaded region**: The inequality says \(y > \frac{2}{3}x - 4\), which means we shade the region **above** the line. This corresponds to everything **to the left** of the line because of how the line is oriented with a positive slope.

### Step 3: Choose the correct graph
The correct graph should have:
- A dashed line with a positive slope, going through \((0, -4)\) and \((3, -2)\).
- The shading on the **left** side of the line (above the line for \(y > \)).

Therefore, the correct option is:
**"On a coordinate plane, a dashed straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the left of the line is shaded."**

Do you want more details or have any questions?

Here are some related questions:
1. How would the graph change if the inequality was \(2x - 3y \geq 12\)?
2. What happens if the inequality was flipped to \(2x - 3y > 12\)?
3. How do you graph an inequality with a negative slope?
4. What is the difference between shading above or below the line for an inequality?
5. How can you determine which side of the line to shade for any inequality?

**Tip**: Always flip the inequality sign when dividing or multiplying by a negative number during your solving process.

Which is the graph of the linear inequality 2x – 3y < 12?

Explore how to graph the linear inequality 2x – 3y < 12. This solution covers transforming the inequality into slope-intercept form and identifying the dashed boundary line with positive slope. Learn how to correctly shade the solution region and apply key inequality rules. Suitable for students in Grades 8-10.

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