Summary systems of Linear Inequalities in Two Variables
- Linear inequalities in two variables describe relationships where one algebraic expression is compared to another using inequality symbols (e.g., $(<)$, $(>)$, $(\leq)$, $(\geq)$).
- A system of linear inequalities consists of two or more inequalities with two variables, typically (x) and (y).
- The solution to the system is the set of ordered pairs that satisfy all inequalities simultaneously.
- Graphically, each inequality divides the plane into two regions: one that satisfies the inequality and one that does not.
- The solution to the system is the region where the shaded areas of all inequalities overlap. The boundaries are represented by straight lines, which may be solid if the inequality is $(\leq)$ or $(\geq)$ or dashed if the inequality is $(<)$ or $(>)$.
- Plotting the Inequality:
- Start by graphing the boundary line on a coordinate plane. This line is derived from the linear equation associated with the inequality.
- Use a dashed line for strict inequalities ($(<)$, $(>)$) and a solid line for non-strict inequalities ($(\leq)$, $(\geq)$).
- Determining the Shaded Region:
- Identify which side of the boundary line to shade based on the inequality symbol.
- For inequalities of the form $(>)$ or $(\geq)$, shade above the line; for $(<)$ or $(\leq)$, shade below the line.
- Include the boundary line in the shading if the inequality is $(\geq)$ or $(\leq)$.
- Testing a Point:
- Choose a test point not on the boundary line and substitute its coordinates into the original inequality.
- If the inequality holds true for the test point, the shaded region is correct. If not, adjust the shading accordingly.
- Labeling the Solution Region:
- Clearly label the shaded region as the solution set for the inequality. This shaded area represents all possible solutions that satisfy the inequality.