The image you provided outlines a step-by-step process for solving a system of inequalities. Let's go through the content in the image to help you understand and complete it:

Steps to Solve the System of Inequalities:

1. Put the system of inequalities in general form:

   • Given: $5x + y \leq 3$ $x \geq 2$

2. Write one inequality in slope-intercept form (if applicable):

   • For $5x + y \leq 3$, solve for $y$: $y \leq -5x + 3$
   • This form helps with graphing by showing the slope and y-intercept.

3. Graph the inequalities on a coordinate plane:

   • For the inequality $y \leq -5x + 3$:
     • Plot the line $y = -5x + 3$ as a solid line because the inequality includes "equal to" ($\leq$).
     • Shade below the line since $y$ is less than or equal to the line.
   • For the inequality $x \geq 2$:
     • Plot the vertical line $x = 2$ as a solid line.
     • Shade to the right of the line since $x$ is greater than or equal to 2.

4. Find the critical values in the shaded region:

   • Identify where the shaded regions overlap. This overlapping area is the solution to the system of inequalities.

5. Label the area of interest on the graph:

   • Clearly indicate the feasible region that satisfies both inequalities.

6. Check a point in the solution region:

   • Verify that the point satisfies both inequalities.

Questions Related to This Topic:

1. How can you determine if a line should be dashed or solid when graphing inequalities?
2. What does the slope-intercept form of a line reveal about the line?
3. How can you verify the solution region of a system of inequalities?
4. What is the significance of shading in graphing inequalities?
5. How do you handle systems of inequalities with more than two inequalities?

Tip:

Always double-check your shaded regions when graphing inequalities to ensure that you have the correct overlap for your solution set.