Systems of Inequalities in Two Variables
What is a System of Inequalities?
A system of inequalities is a set of two or more inequalities with the same variables, considered simultaneously. The solution to the system is the set of all ordered pairs $(x, y)$ that satisfy every inequality in the system at the same time.
Formal Definition: A system of linear inequalities in two variables is a set of inequalities of the form:
$$ \begin{cases} a_1x + b_1y \leq c_1 \\ a_2x + b_2y \leq c_2 \end{cases} $$
where the inequality symbols can be $\leq$, $\geq$, $<$, or $>$. The solution set is the intersection (overlap) of all the individual solution regions.
Key Concepts to Remember
1. Boundary Lines
- Each inequality has a boundary line — the equation you get when you replace the inequality symbol with $=$.
- If the inequality is $\leq$ or $\geq$, draw a solid line (points on the line ARE included).
- If the inequality is $<$ or $>$, draw a dashed line (points on the line are NOT included).
2. Shading Regions
- Each inequality divides the coordinate plane into two half-planes.
- Use a test point (often $(0,0)$ if it's not on the line) to determine which side to shade.
- If the test point satisfies the inequality, shade the side containing the test point.
3. Solution Region (Feasible Region)
- The solution to the system is the overlapping region where ALL shaded areas intersect.
- Every point in this region satisfies all inequalities simultaneously.
4. Corner Points (Vertices)
- The points where boundary lines intersect are called vertices or corner points.
- These are found by solving the corresponding equations as a system of equations.