SYSTEM OF INEQUALITIES

System Of Inequalities

Definition Of System Of Inequalities

A System of Equations is a set of two or more equations with the same variables graphed on the same coordinate plane.

Examples of System of Inequalities

The figure below shows the graph of the system of inequalities
2x - 6y < 12,="" 3x="" +="" 4y="">< 12,="" and="" 4x="" +="" 2y="" ≥="" 8.="">

Video Examples: Algebra 2- Solving Systems of Equations

Solved Example on System of Inequalities

Ques: Which of the graphs defines the system of inequalities, x + y < 3,="" -="" 5x="" +="" 2y="" ≤="" 10,="" y="" ≥="" -="">

Choices:

A. Graph-1
B. Graph-2
C. Graph-3
D. Graph-4
Correct Answer: A

Solution:

Step 1: Graph all the three inequalities in the same coordinate plane.

Step 2: The graph of x + y < 3="" is="" the="" half-plane="" below="" the="" dashed="" line="" x="" +="" y="3.">
Step 3: The graph of - 5x + 2y ≤ 10 is the half-plane on and below the solid line - 5x + 2y = 10.
Step 4: The graph of y ≥ - 3 is the half-plane on and above the solid line y = - 3.
Step 5: The graph of the system is the intersection of the three half planes as shown in the graph.
Step 6: Graph-1 is the correct answer.

Quick Summary

A system of inequalities consists of two or more inequalities.
The solution to a system of inequalities is the set of all points that satisfy all inequalities in the system.
Graphically, the solution is the intersection of the shaded regions of each inequality.

\[ null \]

🍎 Teacher Insights

Emphasize the importance of accurate graphing. Encourage students to test points to verify the shaded region. Provide various examples, including those with no solutions or infinitely many solutions.

🎓 Prerequisites

Solving linear equations
Graphing linear equations
Understanding inequalities
Coordinate plane

Check Your Knowledge

Q1: Which graph represents the solution to the system of inequalities: y > x + 1, y < -x + 3?

A. Graph with overlapping region above y=x+1 and below y=-x+3 B. Graph with overlapping region below y=x+1 and above y=-x+3 C. Graph with no overlapping region D. A single line

Q2: Does the point (0, 0) satisfy the system of inequalities: x + y < 2, x - y > 1?

A. Yes B. No C. Only the first inequality D. Only the second inequality

Frequently Asked Questions

Q: How do you know whether to use a solid or dashed line when graphing an inequality?
A: Use a solid line for inequalities with ≤ or ≥ (inclusive). Use a dashed line for inequalities with < or > (exclusive).

Q: How do you determine which side of the line to shade?
A: Choose a test point (not on the line) and substitute its coordinates into the inequality. If the inequality is true, shade the side containing the test point. If it's false, shade the other side.

Q: What does it mean if a system of inequalities has no solution?
A: It means there is no region where all the inequalities are simultaneously satisfied. Graphically, the shaded regions of the inequalities do not overlap.