Graph
of an Equation or Inequality
The picture obtained by plotting all the points of an equation or inequality.
If there is only one variable,
the graph is on a number line.
If there are two variables, the graph is on the coordinate
plane. If there are three variables,
the graph is in three-dimensional
coordinates. In general, for n variables,
the graph is in n dimensions.
For example, the graph of the equation y
= x + 1
given below is obtained by plotting all the points with a y-coordinate
that is
1 larger
than the
x-coordinate.
See
also
Function, relation, coordinates
Worked Example
Problem: Graph the equation y = 2x − 3 by finding and plotting key points.
Step 1: Choose several values of x. Pick x = 0, 1, 2, and 3.
Step 2: Substitute each x-value into the equation to find y.
x=0:y=2(0)−3=−3x=1:y=2(1)−3=−1x=2:y=2(2)−3=1x=3:y=2(3)−3=3 Step 3: List the ordered pairs: (0, −3), (1, −1), (2, 1), (3, 3).
Step 4: Plot these four points on the coordinate plane. Since the equation is linear (degree 1), connect them with a straight line extending in both directions.
Step 5: Verify: every point on this line has a y-coordinate equal to twice its x-coordinate minus 3. The line is the complete graph of the equation.
Answer: The graph is a straight line passing through (0, −3) and (2, 1) with slope 2, extending infinitely in both directions.
Another Example
This example shows an inequality rather than an equation. The graph is a shaded region instead of a single curve, and the boundary line style (dashed vs. solid) matters.
Problem: Graph the inequality y > x + 1 on the coordinate plane.
Step 1: First, graph the boundary equation y = x + 1. Find two points: when x = 0, y = 1 and when x = 3, y = 4. Plot (0, 1) and (3, 4).
Step 2: Because the inequality is strict (>, not ≥), draw the boundary line as a dashed line. A dashed line means points on the line itself are NOT included in the solution.
Step 3: Pick a test point not on the line—(0, 0) is convenient. Substitute into the inequality.
0>0+1⇒0>1(false) Step 4: Since (0, 0) does not satisfy the inequality, shade the opposite side of the line—the region above the line where y-values are greater than x + 1.
Answer: The graph is the entire region above the dashed line y = x + 1. Every point in the shaded region satisfies y > x + 1.
Frequently Asked Questions
What is the difference between the graph of an equation and the graph of an inequality?
The graph of an equation consists of only the points whose coordinates make the equation true—typically a curve or line. The graph of an inequality includes an entire region of points that satisfy the inequality, shown by shading one side of the boundary curve. The boundary itself is solid if the inequality includes equality (≤ or ≥) and dashed if it does not (< or >).
How do you know if a point is on the graph of an equation?
Substitute the point's coordinates into the equation. If both sides are equal after substitution, the point is on the graph. For example, to test whether (2, 5) is on the graph of y = 2x + 1, compute 2(2) + 1 = 5, which equals the y-coordinate, so the point is on the graph.
How many points do you need to graph an equation?
It depends on the type of equation. A linear equation (degree 1) requires at least two points to determine the line, though plotting a third as a check is wise. A quadratic equation needs at least three points, including the vertex. More complex curves may require many points or knowledge of the equation's shape.
Graph of an Equation vs. Graph of an Inequality
| Graph of an Equation | Graph of an Inequality |
|---|
| What you plot | All points (x, y) that make the equation true | All points (x, y) that make the inequality true |
| Result | A curve or line (one-dimensional on a plane) | A shaded region (two-dimensional area on a plane) |
| Boundary | The curve itself is the entire graph | The boundary curve may or may not be included |
| Line style | Always a solid line or curve | Solid line for ≤ or ≥; dashed line for < or > |
| Test point needed? | No—just plot points satisfying the equation | Yes—use a test point to determine which side to shade |
Why It Matters
Graphing equations and inequalities is one of the most fundamental skills in algebra and precalculus. You use it to visualize solutions, find intersections between functions, and solve systems of equations or inequalities. In real-world applications, graphs let you model and interpret relationships—such as cost versus quantity, distance versus time, or constraints in optimization problems.
Common Mistakes
Mistake: Using a solid line for strict inequalities (< or >).
Correction: A strict inequality means points on the boundary are NOT solutions. Always draw a dashed line for < or > and a solid line only for ≤ or ≥.
Mistake: Plotting too few points and assuming the shape of a nonlinear graph.
Correction: Two points are enough only for a straight line. For parabolas, circles, or other curves, plot several points—including key features like the vertex, intercepts, or turning points—to capture the true shape.
