https://file.aiquickdraw.com/mathbot/file/1726456077202gt8vtnqe.jpg

The inequality you are asked to graph is:

\[
5x + 6y < 42
\]

To graph this inequality, follow these steps:

### Step 1: Convert the inequality to an equation
We start by converting the inequality into an equation to find the boundary line:

\[
5x + 6y = 42
\]

### Step 2: Find the intercepts of the boundary line
- **X-intercept:** Set \( y = 0 \) and solve for \( x \):
  \[
  5x + 6(0) = 42 \implies 5x = 42 \implies x = \frac{42}{5} = 8.4
  \]
  So, the x-intercept is \( (8.4, 0) \).

- **Y-intercept:** Set \( x = 0 \) and solve for \( y \):
  \[
  5(0) + 6y = 42 \implies 6y = 42 \implies y = \frac{42}{6} = 7
  \]
  So, the y-intercept is \( (0, 7) \).

### Step 3: Draw the boundary line
Plot the points \( (8.4, 0) \) and \( (0, 7) \), then draw a dashed line through them. The line will be dashed because the inequality is strict (\(<\)).

### Step 4: Determine the region to shade
Since the inequality is \( 5x + 6y < 42 \), the region below the boundary line (the side where \( y \) is smaller) will be shaded.

### Final Graph:
- A dashed line from \( (8.4, 0) \) to \( (0, 7) \).
- The region below this line will be shaded to represent the inequality \( 5x + 6y < 42 \).

Would you like further details or another explanation?

### Related Questions:
1. How do you solve systems of inequalities graphically?
2. What does it mean if an inequality has a “greater than or equal to” sign?
3. How can you verify a solution for a system of inequalities?
4. How does graphing an inequality differ from graphing an equation?
5. What happens to the graph if we multiply the inequality by a negative number?

### Tip:
When graphing inequalities, always pay attention to whether the inequality includes \( \leq \) or \( \geq \), as this affects whether the boundary line is solid or dashed!

Learn how to graph the linear inequality 5x + 6y < 42 step by step. This problem includes finding intercepts, drawing boundary lines, and shading the appropriate region. Suitable for students in Grades 9-10.

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