7.6 systems of inequalities worksheet | PDF - Free Printable

Problem Analysis:

The worksheet involves two main tasks:
1. Identifying solutions to a system of inequalities (based on the graph provided).
2. Graphing individual inequalities.

Let's solve each part step by step.

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Part 1: Identifying Solutions to the System of Inequalities

#### Given Graph:
The graph shows a shaded region that represents the solution set of a system of inequalities. The boundary lines are drawn, and the shaded area indicates where all inequalities are satisfied simultaneously.

#### Points to Check:
We need to determine whether the following points lie within the shaded region:
1. \( (0, 0) \)
2. \( (-1, 4) \)
3. \( (2, 5) \)
4. \( (5, -1) \)
5. \( (3, 2) \)

#### Solution:
To check if a point is a solution, we simply verify whether it lies within the shaded region:
- Point (0, 0): This point is outside the shaded region.
- Point (-1, 4): This point is outside the shaded region.
- Point (2, 5): This point is inside the shaded region.
- Point (5, -1): This point is outside the shaded region.
- Point (3, 2): This point is inside the shaded region.

#### Final Answer for Part 1:
The points that are solutions to the system of inequalities are:
- \( (2, 5) \)
- \( (3, 2) \)

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Part 2: Solving Each Inequality

#### Problem 6: \( x > 5 \), \( y \leq 4 \)

1. Graph \( x > 5 \):
- Draw a vertical dashed line at \( x = 5 \) (dashed because the inequality is strict).
- Shade the region to the right of this line (where \( x > 5 \)).

2. Graph \( y \leq 4 \):
- Draw a horizontal solid line at \( y = 4 \) (solid because the inequality includes equality).
- Shade the region below this line (where \( y \leq 4 \)).

3. Combine the Shaded Regions:
- The solution is the intersection of the two shaded regions: above \( x = 5 \) and below \( y = 4 \).

#### Problem 7: \( y < 0 \), \( x \geq 0 \)

1. Graph \( y < 0 \):
- Draw a horizontal dashed line at \( y = 0 \) (dashed because the inequality is strict).
- Shade the region below this line (where \( y < 0 \)).

2. Graph \( x \geq 0 \):
- Draw a vertical solid line at \( x = 0 \) (solid because the inequality includes equality).
- Shade the region to the right of this line (where \( x \geq 0 \)).

3. Combine the Shaded Regions:
- The solution is the intersection of the two shaded regions: below \( y = 0 \) and to the right of \( x = 0 \).

#### Problem 8: \( y < -3 \), \( x - y > 1 \)

1. Graph \( y < -3 \):
- Draw a horizontal dashed line at \( y = -3 \) (dashed because the inequality is strict).
- Shade the region below this line (where \( y < -3 \)).

2. Graph \( x - y > 1 \):
- Rewrite the inequality as \( y < x - 1 \).
- Draw the line \( y = x - 1 \) (dashed because the inequality is strict).
- Shade the region below this line (where \( y < x - 1 \)).

3. Combine the Shaded Regions:
- The solution is the intersection of the two shaded regions: below \( y = -3 \) and below \( y = x - 1 \).

#### Problem 9: \( y < 2x + 3 \), \( y < -x + 1 \)

1. Graph \( y < 2x + 3 \):
- Draw the line \( y = 2x + 3 \) (dashed because the inequality is strict).
- Shade the region below this line (where \( y < 2x + 3 \)).

2. Graph \( y < -x + 1 \):
- Draw the line \( y = -x + 1 \) (dashed because the inequality is strict).
- Shade the region below this line (where \( y < -x + 1 \)).

3. Combine the Shaded Regions:
- The solution is the intersection of the two shaded regions: below both \( y = 2x + 3 \) and \( y = -x + 1 \).

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Final Answers:

1. Points that are solutions to the system of inequalities:
- \( \boxed{(2, 5), (3, 2)} \)

2. Graphs for each inequality:
- Problem 6: Shaded region to the right of \( x = 5 \) and below \( y = 4 \).
- Problem 7: Shaded region below \( y = 0 \) and to the right of \( x = 0 \).
- Problem 8: Shaded region below \( y = -3 \) and below \( y = x - 1 \).
- Problem 9: Shaded region below \( y = 2x + 3 \) and below \( y = -x + 1 \).