Graphing Linear Inequalities
Graphing linear inequalities is a Grade 7 math skill from Yoshiwara Intermediate Algebra where students graph a boundary line, choose solid or dashed based on the inequality symbol, then shade the correct half-plane. The shaded region represents all ordered pairs that satisfy the inequality.
Key Concepts
Property A linear inequality can be written in the form $$ax + by + c \leq 0 \quad \text{or} \quad ax + by + c \geq 0$$ The solutions consist of a boundary line and a half plane . If the inequality is equivalent to $y \geq mx + b$, shade the half plane above the line. If it is equivalent to $y \leq mx + b$, shade below the line. An inequality with $ $ or $<$ is strict , and its boundary line is dashed.
Examples To graph $3x y 6$, we first solve for $y$. This gives $ y 3x + 6$, which becomes $y < 3x 6$ after dividing by $ 1$. We draw a dashed line for $y = 3x 6$ and shade the half plane below it.
To graph $x + 2y \leq 8$, we solve for $y$ to get $2y \leq x + 8$, or $y \leq \frac{1}{2}x + 4$. We draw a solid line for $y = \frac{1}{2}x + 4$ and shade the region below it.
Common Questions
How do you graph a linear inequality?
Graph the boundary line (solid for ≤/≥, dashed for </>) in the same way as the equation. Then use a test point to determine which side to shade.
When do you use a dashed vs solid boundary line?
Use a solid line for ≤ or ≥ (boundary is included). Use a dashed line for < or > (boundary is not included in the solution).
How do you determine which half-plane to shade?
Test a point not on the line. If it satisfies the inequality, shade the region containing it. If not, shade the other region.
What does the shaded region in a linear inequality represent?
The shaded region represents all (x, y) pairs that make the inequality true — the solution set of the inequality.