To graph the inequality $x + y \geq 10$, follow these steps:

1. Rewrite the inequality in slope-intercept form:

Solve for $y$ in terms of $x$: $x + y \geq 10 \quad \Rightarrow \quad y \geq -x + 10$ This shows the boundary line $y = -x + 10$ and the inequality $y \geq -x + 10$.

2. Plot the boundary line:

• The boundary line is $y = -x + 10$.
• Find two points on the line:
  • When $x = 0$, $y = 10$ (point: $(0, 10)$).
  • When $y = 0$, $x = 10$ (point: $(10, 0)$).
• Plot these points on the graph and draw a solid line (because $\geq$ includes equality).

3. Shade the solution region:

• Since $y \geq -x + 10$, shade the region above the line (where $y$ values are greater than or equal to $-x + 10$).

4. Final graphing:

• The shaded area includes the boundary line and all points above it.

Would you like additional help with graphing tools, or do you want me to explain any specific part of the process?

Here are 5 questions to deepen your understanding:

1. How can you verify if a point lies within the solution set of the inequality?
2. What changes in the graph if the inequality is $x + y < 10$ instead of $x + y \geq 10$?
3. How would the graph change if the inequality were $x + y \leq 10$?
4. What does the slope of the boundary line represent in this inequality?
5. How can we apply similar graphing methods to systems of inequalities?

Tip:

Always use test points (like $(0, 0)$) to verify the shaded region matches the inequality!