Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set. y < -4/5 x + 7 and y > 2x - 7.

Steps to Solve

1. Identify the inequalities

Start with the given inequalities. For example, let's say we have two inequalities:

1. $y > 2x + 1$

2. $y \leq -x + 4$

3. Graph the first inequality

To graph $y > 2x + 1$, first start by graphing the line $y = 2x + 1$. This line has a slope of 2 and a y-intercept of 1.

• Draw a dashed line since the inequality is “greater than” (not including the line).
• Next, shade the area above the line to indicate that we're interested in points where $y$ is greater than $2x + 1$.

3. Graph the second inequality

Now graph the second inequality $y \leq -x + 4$.

• Start by graphing the line $y = -x + 4$. This line has a slope of -1 and a y-intercept of 4.
• This time, draw a solid line because the inequality includes equality (less than or equal to).
• Shade the area below this line to show that we're focused on points where $y$ is less than or equal to $-x + 4$.

4. Identify the feasible region

The feasible region is the area where the shaded areas from both inequalities overlap. This is the solution to the system of inequalities.

5. Check a point within the feasible region

Choose a point within the shaded overlap area to determine if it satisfies both inequalities. For example, you might choose the point (0, 3).

• Check it against both inequalities:
  • For $y > 2x + 1$: $3 > 2(0) + 1$ (True)
  • For $y \leq -x + 4$: $3 \leq -(0) + 4$ (True)

Since the point satisfies both inequalities, it is a valid solution.