Graphing Linear Inequalities — Definition, Formula & Examples
Graphing linear inequalities means representing an inequality like on the coordinate plane by drawing a boundary line and shading the region of all points that make the inequality true.
A linear inequality in two variables has the form , , , or . Its graph is a half-plane: the set of all ordered pairs satisfying the inequality, bounded by the line , which is drawn solid when the inequality includes equality ( or ) and dashed when it does not ( or ).
Key Formula
Where:
- = Coefficients of the variables (not both zero)
- = Variables representing coordinates on the plane
- = Constant on the right side of the inequality
How It Works
To graph a linear inequality, first graph the boundary line by replacing the inequality symbol with an equals sign. Use a solid line for or and a dashed line for or . Then pick a test point not on the line — the origin is the easiest choice when it is not on the boundary. Substitute that point into the original inequality: if the result is true, shade the side of the line containing the test point; if false, shade the opposite side. Every point in the shaded region is a solution to the inequality.
Worked Example
Problem: Graph the inequality on the coordinate plane.
Step 1: Draw the boundary line: Replace the inequality with an equals sign and graph the line . The y-intercept is and the slope is , so another point is . Because the symbol is (includes equality), draw a solid line.
Step 2: Choose a test point: Pick the origin since it does not lie on the boundary line.
Step 3: Substitute into the inequality: Plug into .
Step 4: Shade the correct region: The test point makes the inequality true, so shade the side of the line that contains the origin — the region below and to the left of the line.
Answer: Draw a solid line through and , then shade below it. Every point in the shaded region satisfies .
Another Example
Problem: Graph the inequality .
Step 1: Rewrite in slope-intercept form (optional but helpful): Solve for : subtract and divide by . Remember to flip the inequality when dividing by a negative number.
Step 2: Graph the boundary line: Graph using the y-intercept and slope , giving a second point . The symbol is (strict), so draw a dashed line.
Step 3: Test the origin: Substitute into the original inequality .
Step 4: Shade the opposite side: Since the origin does not satisfy the inequality, shade the region on the other side of the line — below and to the right.
Answer: Draw a dashed line through and , then shade below it. All points in the shaded region satisfy .
Why It Matters
Graphing linear inequalities is a core skill in Algebra 1 and appears again in Algebra 2 when you solve systems of inequalities to find feasible regions. In economics and business, linear programming uses these shaded regions to optimize profit, cost, or resource allocation under constraints.
Common Mistakes
Mistake: Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
Correction: When you multiply or divide both sides of an inequality by a negative value, you must reverse the direction of the inequality symbol. For example, dividing by gives , not .
Mistake: Shading the wrong side of the boundary line without testing a point.
Correction: Always substitute a test point (such as the origin) into the original inequality to confirm which half-plane to shade. Do not assume 'greater than' always means shade above.