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To graph the inequality \( 3x + 4y < 12 \), we will follow these steps:

### 1. Convert the inequality to an equation
We first turn the inequality into the equation:
\[
3x + 4y = 12
\]
This will give us the boundary line for the inequality.

### 2. Find the intercepts
To graph the line, it's easiest to find the x- and y-intercepts by setting one variable to zero at a time:

- **x-intercept**: Set \( y = 0 \):
  \[
  3x + 4(0) = 12 \implies x = 4
  \]
  So the x-intercept is \( (4, 0) \).

- **y-intercept**: Set \( x = 0 \):
  \[
  3(0) + 4y = 12 \implies y = 3
  \]
  So the y-intercept is \( (0, 3) \).

### 3. Plot the boundary line
On the graph, plot the points \( (4, 0) \) and \( (0, 3) \). These points will give you the boundary line for \( 3x + 4y = 12 \). Since the inequality is \( < \) (strictly less than), the boundary line should be **dashed** to indicate that points on the line are not included in the solution.

### 4. Shade the appropriate region
To determine which side of the line to shade, pick a test point (usually \( (0, 0) \) unless it's on the line). Substituting \( (0, 0) \) into the inequality:
\[
3(0) + 4(0) < 12 \implies 0 < 12
\]
This is true, so we shade the region that contains \( (0, 0) \).

### Final Graph
- The boundary line will be dashed.
- The region below the line, including the origin, will be shaded.

Would you like further clarification on any step or need help drawing the graph?

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Here are 5 related questions:
1. How would the graph change if the inequality were \( 3x + 4y \leq 12 \)?
2. How do we determine the test point for shading when the origin lies on the line?
3. How can you find the slope of the boundary line from the equation?
4. What does the solution set of this inequality represent in a real-world context?
5. How would the graph change if the inequality were \( 3x + 4y > 12 \)?

**Tip:** Always use a dashed line for inequalities with \( < \) or \( > \), and a solid line for \( \leq \) or \( \geq \).

Graph the inequality 3x + 4y < 12 on the axes below.

Learn how to graph the inequality 3x + 4y < 12 by finding intercepts, plotting the boundary line, and shading the correct region. This solution includes step-by-step instructions for identifying the intercepts and determining which region of the graph satisfies the inequality.

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