Linear Inequalities in Two Variables
📌 What is a Linear Inequality in Two Variables?
A linear inequality in two variables has one of the following forms:
$ax + by < c$ or $ax + by > c$ or $ax + by \leq c$ or $ax + by \geq c$
where $x$ and $y$ are variables, and $a$, $b$, $c$ are constants (real numbers), with $a$ and $b$ not both zero.
Key Idea: Unlike a linear equation (which gives a line), a linear inequality gives an entire region (half-plane) of the coordinate plane as its solution set.
✅ What is a Solution?
A solution of a linear inequality in two variables is an ordered pair $(x, y)$ that makes the inequality true when the values are substituted in.
✓ Example: A Solution
Is $(1, 2)$ a solution of $0.5x^2 - 2y + 1 < 0$?
Substitute $x = 1$, $y = 2$:
$$0.5(1)^2 - 2(2) + 1 = 0.5 - 4 + 1 = -2.5$$
Since $-2.5 < 0$ is TRUE, $(1, 2)$ is a solution. ✅
✗ Example: Not a Solution
Is $(2, 1)$ a solution of $0.5x^2 - 2y + 1 < 0$?
Substitute $x = 2$, $y = 1$:
$$0.5(2)^2 - 2(1) + 1 = 2 - 2 + 1 = 1$$
Since $1 > 0$ (NOT $< 0$), $(2, 1)$ is NOT a solution. ❌