To solve the system of inequalities by graphing, we'll graph each inequality and find the region where the solutions overlap. Here are the steps:

Inequality 1: $y - x < 3$

1. Rewrite the inequality in slope-intercept form (i.e., $y = mx + b$): $y < x + 3$
2. Graph the boundary line $y = x + 3$. This line will have a slope of 1 and a y-intercept of 3.
3. Determine the region to shade: Since the inequality is $y < x + 3$, shade the region below the line.

Inequality 2: $y - x \geq 2$

1. Rewrite the inequality in slope-intercept form: $y \geq x + 2$
2. Graph the boundary line $y = x + 2$. This line will also have a slope of 1 and a y-intercept of 2.
3. Determine the region to shade: Since the inequality is $y \geq x + 2$, shade the region above the line.

Graphing and Finding the Solution

1. Graph both boundary lines:
   • $y = x + 3$ (dashed line, because the inequality is strict)
   • $y = x + 2$ (solid line, because the inequality is inclusive)
2. Shade the appropriate regions:
   • Below the line $y = x + 3$
   • Above the line $y = x + 2$
3. Identify the overlapping region: The solution set is the region where the shaded areas overlap.

Solution

The solution is the region between the lines $y = x + 2$ and $y = x + 3$, where:

• $y > x + 2$ (below the dashed line)
• $y \leq x + 3$ (above the solid line)

This region forms a strip parallel to the lines, above $y = x + 2$ and below $y = x + 3$.

Would you like me to graph this for you or provide further details?

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Here are 5 related questions:

1. How do you solve systems of linear inequalities with more than two inequalities?
2. What is the difference between a strict inequality and an inclusive inequality?
3. How does the slope of a line affect the solution of a system of inequalities?
4. What happens when the inequalities in a system have no overlapping region?
5. Can you explain how to find the vertices of the solution region for a system of linear inequalities?

Tip: When graphing inequalities, always pay attention to whether the boundary line should be dashed or solid—this indicates whether the line itself is part of the solution set.