To graph the inequality $x - y < 3$, we can follow these steps:

Step 1: Rewrite the inequality in slope-intercept form (if possible)

We want to isolate $y$ on one side. Start by subtracting $x$ from both sides of the inequality:

$-x + x - y < 3 - x$

$-y < -x + 3$

Now multiply both sides by -1. Remember, when multiplying or dividing by a negative number, the inequality sign flips:

$y > x - 3$

So, the inequality is now $y > x - 3$.

Step 2: Graph the boundary line

The equation $y = x - 3$ is the boundary line of the inequality. This is a line with:

• Slope = 1 (since the coefficient of $x$ is 1),
• Y-intercept = -3 (since the constant term is -3).

To graph this line, plot the y-intercept at $(0, -3)$, then use the slope (rise over run) of 1 to move up 1 unit and 1 unit to the right, plotting another point at $(1, -2)$. Draw a straight line through these points.

Step 3: Determine which side of the line to shade

The inequality is $y > x - 3$, meaning the solution consists of all points above the line $y = x - 3$.

To confirm, you can pick a test point that is not on the line. A good test point is the origin $(0, 0)$. Substitute $x = 0$ and $y = 0$ into the inequality:

$0 > 0 - 3$

$0 > -3 \quad \text{(True)}$

Since the test point satisfies the inequality, we shade the region above the line.

Final Graph:

1. The boundary line $y = x - 3$ is drawn as a dashed line because the inequality is strict ($<$, not $\leq$).
2. The shaded region will be above this line.

Would you like to see a visual representation of the graph or have any other questions about graphing inequalities?

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Here are five related questions you might find interesting:

1. How do you graph inequalities involving $\leq$ or $\geq$?
2. What if the inequality was $x + 2y > 4$? How would the graph change?
3. How do you graph inequalities in two variables that have an "and" condition (e.g., $y < 3x + 1$ and $y > x - 2$)?
4. How can you determine the solution set for a system of linear inequalities?
5. What would happen if the inequality was $y \leq x - 3$ instead?

Tip: When graphing inequalities, always check if the inequality is strict (using a dashed line) or non-strict (using a solid line) before shading!