Lesson 43: Solving Systems of Linear Inequalities

New Concept

A system of linear inequalities is formed by two or more linear inequalities.

What’s next

Next, you'll graph these systems to find the solution—the intersection of the shaded half-planes for each inequality.

A system of linear inequalities is formed by two or more linear inequalities. An ordered pair is a solution of a system if it is a solution of every inequality in the system.

Is $(2, 1)$ a solution for the system $y < 3$ and $x + 2y \leq 4$? Yes! Because $1 < 3$ is true AND $2 + 2(1) \leq 4$ is true.
Is $(2, 2)$ a solution for the same system? Nope! While $2 < 3$ is true, $2 + 2(2) \leq 4$ is false. It failed one of the rules!

Think of it as getting into a secret club with multiple rules! To be a member, you must follow every single rule (inequality) perfectly. If you break even one, you're out. The graph's solution is the cool, overlapping shaded area where all the 'rule zones' meet. It’s the spot where every condition is satisfied at once.

Property

When drawing the boundary line of a linear inequality, you must use specific formatting to show whether the line itself is part of the solution set:

• Dashed Line: Use for strictly "less than" ($<$) or "greater than" ($>$). Points sitting exactly on a dashed line are NOT solutions.

• Solid Line: Use for "less than or equal to" ($\leq$) or "greater than or equal to" ($\geq$). Points sitting exactly on a solid line ARE solutions.

Property

When the boundary lines of a system are parallel (they have the exact same slope but different y-intercepts), the shading dictates whether a solution exists:

• No Solution: If the shaded regions face away from each other and never overlap, there is no solution ($\emptyset$).
• Strip Region (Infinite Solutions): If the shaded regions face inward toward each other, the solution is the entire parallel "lane" trapped between the two boundary lines.

Examples

• Example 1 (No Solution): Solve the system $y > x + 3$ and $y < x - 1$.

The first inequality shades everything ABOVE the higher parallel line. The second shades everything BELOW the lower parallel line. Because the regions point away from each other, they will never cross. There is No Solution.

• Example 2 (Strip Region): Solve the system $y \leq -x + 5$ and $y \geq -x + 1$.

The first inequality shades BELOW the higher parallel line. The second shades ABOVE the lower parallel line. Because they shade inward toward each other, the solution is the endless parallel strip of space trapped between the two lines.

Explanation

Parallel lines usually mean "no solution" when we are dealing with standard equations (because the lines never touch). But with inequalities, we are dealing with massive shaded areas! Imagine two parallel highways. If one rule says "shade everything North of Highway A", and the other says "shade everything South of Highway B", the shadings will never meet (No Solution). But if the rules shade the space in between the two highways, you end up with a massive, endless river of solutions flowing between them!