How to Graph Inequalities and Systems
An equation has a solution set you can draw as a line. An inequality has a solution set that fills a region. Everything follows from two decisions: is the boundary included, and which side of it is the answer.
Boundary, then side
Graphing any inequality is the same two-step procedure.
- Draw the boundary. Replace the inequality symbol with
=and graph the resulting equation. Make it dashed for<or>, solid for≤or≥. - Shade the correct side. The boundary splits the plane in two; one half satisfies the inequality and the other does not.
The test-point method
When an inequality is written as y < something, the shading
is obvious: below for <, above for >. When
it is not in that form, do not guess. Test a point.
- Pick any point that is clearly not on the boundary.
- Substitute its coordinates into the original inequality.
- If the statement comes out true, shade the side containing that point. If false, shade the other side.
Worked example
Graph 3x - 2y ≥ 6.
The boundary is 3x - 2y = 6, which passes through
(2, 0) and (0, -3). The symbol is ≥,
so the line is solid.
Test the origin: 3(0) - 2(0) = 0, and 0 ≥ 6 is
false. So the origin is not in the solution, and the region on the
other side of the line is shaded.
The origin is almost always the best test point, because the arithmetic is
trivial. The one time you cannot use it is when the boundary passes through
it, in which case pick something like (1, 0).
Non-linear inequalities
Nothing changes. The boundary is just a curve rather than a line, and the region it bounds may be more interesting.
For a circle, x² + y² < 25 is the inside of a circle of
radius 5, and x² + y² > 25 is everything outside it. The
test-point method settles which is which in one substitution: the origin
gives 0 < 25, true, so the inside is the solution.
Systems and feasible regions
A system of inequalities asks for the points satisfying all of them at once. Graph each one, and the answer is the region where every shading overlaps.
Take the system:
y ≤ -x + 6
y ≤ 2x + 3
y ≥ 0
The overlap is a triangle. In linear programming this is the feasible region, and the key result is that the maximum or minimum of any linear objective function occurs at one of its corners. So the procedure is:
- Graph every constraint and identify the feasible region.
- Find the coordinates of each corner, which means solving the pair of equations that meet there.
- Evaluate the objective function at each corner.
- The best value among them is the answer.
The Analyse tab finds those corners for you: each one is an intersection of two boundaries, reported to six figures.
The sign flip everyone forgets
When you multiply or divide both sides of an inequality by a negative number, the symbol reverses.
-2y < 6
Divide both sides by -2:
y > -3
The reason is that negating reverses the order of the number line. It is
easy to verify: 2 < 5 is true, and multiplying both sides by
-1 gives -2 and -5, where -2 is now the larger. The relationship
genuinely reverses, so the symbol must follow.
A free check. After rearranging, pick a value that obviously satisfies your answer and test it in the original inequality. If it fails, you flipped when you should not have, or did not flip when you should. Ten seconds, and it catches the error every time.
Questions
How do I know which side to shade?
Pick any point not on the boundary, substitute it into the inequality, and see whether the statement is true. If it is, shade that side. The origin is easiest whenever the boundary does not pass through it.
When is the boundary line dashed?
When the inequality is strict, using < or >. A dashed line says the boundary itself is not part of the solution. Use a solid line for ≤ and ≥.
What is a feasible region?
The area satisfying every inequality in a system at once. Graphically it is the overlap of all the shaded regions, and in linear programming it is where the optimal solution must lie.
Why does the inequality sign flip when I multiply by a negative?
Because multiplying by a negative reverses the order of the number line. 2 < 5 is true, but multiplying both sides by −1 gives −2 and −5, and −2 is greater than −5. The relationship reverses, so the symbol must too.
Can I graph an inequality that is not solved for y?
Yes. Type it as written and the calculator shades the region where it holds. Rearranging to make y the subject produces a crisper boundary, but it is not required.