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!