6.1 Linear Inequalities in Two Variables
Definition of Linear Inequalities in Two Variables
A linear inequality in two variables is an inequality that involves two variables with an exponent (power) of 1 for each variable. It takes one of the following general forms:
- $ax + by > c$
- $ax + by \ge c$
- $ax + by < c$
- $ax + by \le c$
where $a$, $b$, and $c$ are constants, and $a \ne 0$, $b \ne 0$.
Representing Linear Inequalities Graphically
A linear boundary line $y = mx + c$ divides a Cartesian plane into two distinct regions:
- Solid Line ($\le$ or $\ge$): Indicates that points lying on the line itself are included in the solution region.
- Dashed / Broken Line ($<$ or $>$): Indicates that points lying on the line itself are excluded from the solution region.
Shading the Region Satisfying an Inequality
For an inequality in the form $y \dots mx + c$:
- $y > mx + c$: Shade the region strictly above the dashed line $y = mx + c$.
- $y \ge mx + c$: Shade the region above or on the solid line $y = mx + c$.
- $y < mx + c$: Shade the region strictly below the dashed line $y = mx + c$.
- $y \le mx + c$: Shade the region below or on the solid line $y = mx + c$.
Note: For vertical lines $x = k$, $x > k$ is to the right and $x < k$ is to the left.
6.2 Systems of Linear Inequalities in Two Variables
Definition of a System of Linear Inequalities
A system of linear inequalities is a combination of two or more linear inequalities involving the same set of two variables.
Feasible Region (Solution Region)
The solution region satisfying a system of linear inequalities is the overlapping shaded region that simultaneously satisfies every individual inequality in the system.
Determining Points in the Solution Region
- A point $(x, y)$ belongs to the solution region if substituting its coordinates into all inequalities in the system yields true statements.
- If a point lies on a dashed boundary line, it is not part of the solution region.
- If a point lies on a solid boundary line, it is part of the solution region.
Solving Practical Real-World Problems
Steps to form and solve linear inequalities from situational problems:
- Identify and define the two unknown variables (e.g., $x$ and $y$).
- Translate word constraints into linear inequality expressions using key terms:
- "At least" / "Not less than": $\ge$
- "At most" / "Not more than": $\le$
- "More than" / "Exceeds": $>$
- "Less than": $<$
- Construct the Cartesian plane, plot boundary lines, and identify the feasible common region.