Graphing Inequalities Calculator
Graph a linear inequality in two variables. Enter the coefficients for y [inequality] ax + b, choose <, ≤, >, or ≥, and see which half-plane to shade and whether the boundary is solid or dashed.
Graphing Linear Inequalities Step-by-Step
Start by graphing the boundary line. If the inequality is y ≤ 2x − 3, graph y = 2x − 3. This line has slope 2 and y-intercept −3. Plot the intercept, use the slope to find another point, and draw the line. Because the inequality is ≤, make it a solid line (points on the line count).
Next, determine which side to shade. For y ≤ 2x − 3, shade below the line because y is less than or equal to the expression. Every point in the shaded region, along with the line itself, satisfies the inequality. If the inequality were y < 2x − 3, the line would be dashed (not included), and you'd shade below.
For y > or y ≥, shade above. The shading represents all (x, y) pairs that make the inequality true. Testing a point helps: pick (0, 0) if it's not on the line. Plug it into the inequality. If true, (0, 0) is in the solution region. If false, shade the opposite side.
Understanding Half-Planes
A linear inequality divides the coordinate plane into two half-planes. The boundary line is the division. One half-plane contains all solutions, the other doesn't. Shading highlights the solution half-plane.
The concept extends to absolute value and nonlinear inequalities, but linear ones are simplest. Each inequality y [symbol] mx + b carves out a region. For systems (multiple inequalities), you graph each and find where shadings overlap. The overlapping region satisfies all inequalities.
In optimization (linear programming), you graph constraints as inequalities and find the feasible region. The corners (vertices) of this region are candidates for maximum or minimum values of an objective function. Graphing inequalities turns abstract constraints into visual, geometric problems you can solve by inspection.
Common Mistakes and Tips
Watch the inequality direction when rearranging. If you multiply or divide both sides by a negative number, flip the inequality. For example, −2y < 6 becomes y > −3, not y < −3. Missing the flip is a frequent error.
Remember: solid line for ≤ or ≥, dashed for < or >. A common mistake is using a solid line for strict inequalities. Double-check the symbol before graphing. When in doubt, test a point. Pick an easy one like (0, 0), substitute into the inequality, and see if it works. If it does and (0, 0) is not on the line, shade the side containing (0, 0). If it doesn't, shade the other side.
For systems, graph all inequalities on the same axes. Use different shading patterns (hatching, colors) if needed. The solution is where all patterns overlap. Clearly label each line and inequality to avoid confusion. Practice transforms these steps into automatic, confident graphing.
Frequently Asked Questions
How do you graph a linear inequality?
Graph the boundary line y = ax + b. Use a solid line if the inequality is ≤ or ≥ (the line is part of the solution). Use a dashed line if it's < or > (the line is not included). Then shade the region where the inequality holds: above the line for > or ≥, below for < or ≤.
What's the difference between a solid and dashed line?
A solid line means points on the line satisfy the inequality (≤ or ≥). A dashed line means points on the line do not satisfy the inequality (< or >). The shading shows all other solutions.
How do you know which side to shade?
For y < or y ≤, shade below the line (smaller y values). For y > or y ≥, shade above the line (larger y values). You can also test a point not on the line: if it satisfies the inequality, shade that side; if not, shade the opposite side.
Can you graph inequalities with x and y on both sides?
Yes. Rearrange the inequality to solve for y, getting y < mx + b or similar. Then graph as usual. The rearrangement might flip the inequality sign if you multiply or divide by a negative.
How do you graph a system of inequalities?
Graph each inequality separately, shading its region. The solution to the system is where the shaded regions overlap. This overlapping area satisfies all inequalities simultaneously.