Systems of Linear Inequalities in Two Variables

Section 5.5: Systems of Inequalities

Linear Inequalities in Two Variables

Linear inequalities in two variables are fundamental in College Algebra, allowing us to describe regions of the coordinate plane that satisfy certain conditions. These inequalities can be solved and graphed to visualize their solution sets.

• Definition: A linear inequality in two variables, x and y, is an inequality that can be written as $ax + by < c$, $ax + by > c$, $ax + by \leq c$, or $ax + by \geq c$, where a, b, and c are constants and a and b are not both zero.

• Solution: Any ordered pair $(x, y)$ that satisfies the inequality is a solution.

• Graphing: The graph of a linear inequality is a region of the coordinate plane. The boundary of this region is the line $ax + by = c$.

• Boundary Line: If the inequality is strict ($<$ or $>$), the boundary line is dashed, indicating points on the line are not included in the solution set. If the inequality is inclusive ($\leq$ or $\geq$), the boundary line is solid, indicating points on the line are included.

• Test Point: To determine which side of the boundary line to shade, select a test point (commonly $(0,0)$ if it is not on the line) and substitute its coordinates into the inequality.

• Shading: If the test point satisfies the inequality, shade the side containing the test point. Otherwise, shade the opposite side.

Example: Graph the solution set for $2y - 4x < 16$.

• Rewrite as $2y < 4x + 16$ or $y < 2x + 8$.

• Graph the boundary line $y = 2x + 8$ as a dashed line.

• Test the point $(0,0)$: $0 < 2(0) + 8 \Rightarrow 0 < 8$ (true), so shade the side containing $(0,0)$.

Solving Linear Inequalities in Two Variables

To solve and graph a linear inequality in two variables, follow these steps:

1. Step 1: Replace the inequality symbol with an equals sign to obtain the boundary line.

2. Step 2: Draw the boundary line (solid for $\leq$ or $\geq$, dashed for $<$ or $>$).

3. Step 3: Choose a test point not on the boundary line, substitute into the original inequality, and shade the region that makes the inequality true.

Solving Linear Inequalities Joined by "And" or "Or"

Systems of inequalities may be joined by the logical operators "and" or "or", which affect the solution set:

• Union ("or"): The union of two sets $A$ and $B$, $A \cup B$, is the set containing all elements that are in $A$ or $B$ or both.

• Intersection ("and"): The intersection of two sets $A$ and $B$, $A \cap B$, is the set containing all elements that are in both $A$ and $B$.

• Graphical Representation: The solution set for "and" is the region where the shaded areas of both inequalities overlap. For "or", the solution set is the union of the shaded regions.

Example: Graph the solution set for $5x + 6y < -30$ and $x \geq -4$.

• Graph the boundary line $5x + 6y = -30$ (dashed) and $x = -4$ (solid).

• Shade the region where both inequalities are satisfied (the intersection).

Practice Exercises

• Exercise 1: Graph the solution set of $2y - 4x < 16$.

• Exercise 2: Graph the solution set of $x \leq 3$.

• Exercise 3: Graph the solution set of $y < -1$.

• Exercise 4a: Graph the solution set of $5x + 6y < -30$ and $x \geq -4$.

• Exercise 4b: Graph the solution set of $5x + 6y < -30$ or $x \geq -4$.

Summary Table: Boundary Line Types

Inequality Symbol	Boundary Line Type	Points on Line Included?
$<$ or $>$	Dashed	No
$\leq$ or $\geq$	Solid	Yes

Additional info: The notes provide a concise summary of graphing linear inequalities and systems, including the use of logical operators "and" and "or". The exercises reinforce the concepts with practical graphing problems.