To graph the inequality $-3x + y \leq -3$, we need to follow these steps:

Step 1: Rewrite the inequality in slope-intercept form

The slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. First, solve for $y$:

$-3x + y \leq -3$

Add $3x$ to both sides:

$y \leq 3x - 3$

Step 2: Graph the boundary line

The boundary line corresponds to the equation $y = 3x - 3$. Since the inequality is "less than or equal to," the line will be solid, indicating that points on the line are included in the solution.

1. Slope: $3$ (rise/run = 3/1)
2. Y-intercept: $-3$ (this is where the line crosses the y-axis)

Plot the y-intercept at $(0, -3)$, then use the slope to find another point. From $(0, -3)$, move up 3 units and 1 unit to the right to get the point $(1, 0)$.

Step 3: Shade the region

Since the inequality is $y \leq 3x - 3$, you will shade below the line, because $y$ is less than or equal to the line.

I'll graph this for you:

1. A solid line through $(0, -3)$ and $(1, 0)$.
2. Shaded region below the line.

Let me know if you'd like me to generate a visual representation of the graph!

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Would you like more details on this process? Here are some questions to deepen your understanding:

1. How do you convert inequalities to slope-intercept form?
2. What does the slope of a line represent in a graph?
3. Why do we use a solid line for $\leq$ or $\geq$ inequalities?
4. How do you determine which region to shade in an inequality graph?
5. What would change if the inequality were $y > 3x - 3$?

Tip: Always check if you need to flip the inequality when multiplying or dividing by negative numbers!