How do you solve an equation graphically?

Draw both sides as separate graphs on the same axes. The x-coordinates of the points where they intersect are the solutions.

How do you know which side to shade for an inequality?

Pick a test point not on the boundary, like (0, 0). If it makes the inequality true, shade its side; if false, shade the other side.

When is the boundary line dashed?

When the sign is , because points on the line are not solutions. Use a solid line for ≤ or ≥.

Solving Equations & Inequalities Graphically

The graph of an equation is the set of all its solutions. To solve f(x) = g(x), draw y = f(x) and y = g(x): the x-coordinates where they cross are the solutions. f(x) > g(x) is true for the x-values where the graph of f is above the graph of g. A linear inequality in two variables, like y > 2x + 1, is solved by a half-plane: draw the boundary line (dashed for < or >, solid for ≤ or ≥) and shade the side that a test point says is true. A system of inequalities is solved by the overlap of the shaded half-planes.

The graph is the solution set

Every point (x, y) on the graph of an equation in two variables makes the equation true. Every point off the graph makes it false. So the graph is a picture of all the solutions.

Example: (2, 3) is on y = 2x − 1 because 2 × 2 − 1 = 3. (2, 4) is not, because 3 ≠ 4.

Solving f(x) = g(x) with intersections

To solve an equation like f(x) = g(x):

Draw y = f(x) and y = g(x) on the same axes.
Find where the graphs intersect (cross).
The x-coordinate of each intersection is a solution.

This works for any kinds of functions — linear, quadratic, exponential, absolute value — even when algebra is hard. With a graphing tool or a table of values you can find the answer to any accuracy (successive approximation).

Special case: to solve f(x) = 0, look where y = f(x) crosses the x-axis (y = 0).

Number of solutions = number of crossing points. Parallel lines never cross: no solution.

Solving f(x) > g(x) graphically

f(x) > g(x) is true wherever the graph of f is above the graph of g. f(x) < g(x) is true where f is below.

Find the intersection(s) first: they are the boundary points.
Then look left and right of each boundary point.
For ≥ or ≤, include the intersection point itself.

Example: 2x − 1 > −x + 2. The lines meet at x = 1. To the right, the line y = 2x − 1 is higher, so the answer is x > 1. Check with algebra: 3x > 3, so x > 1. ✔

Linear inequalities in two variables: half-planes

An inequality like y > 2x + 1 or 3x + 2y ≤ 12 is true for a whole region called a half-plane.

Boundary: draw the line you get with '='. Use a dashed line for < or > (points on it are not solutions) and a solid line for ≤ or ≥.
Test point: pick a point not on the line, usually (0, 0). Put it in the inequality.
If it is true, shade the side with the test point. If false, shade the other side.

Shortcut when the inequality is written as y > mx + c: shade above the line; for y < mx + c, shade below.

Systems of linear inequalities

A system has two or more inequalities that must all be true together. Shade each one; the overlap is the solution set (also called the feasible region).

Corner points of the region are found by solving pairs of boundary equations. If the shadings never overlap, the system has no solution. This idea is the base of linear programming, where we find the best point inside the region.

Try it: graph paper check

On graph paper draw y = x + 1 (dashed) and y = −x + 3 (solid). Mark five points: (0, 0), (0, 2), (1, 2), (−1, 3), (2, 0). Before testing, predict which lie in the region y > x + 1 and y ≤ −x + 3. Then test each by substituting. Use the 3D sliders in the last step to check.

Key formulas and definitions

Solution of f(x) = g(x): x-coordinates of intersection points
f(x) > g(x): x-values where graph of f is above graph of g
f(x) = 0: where y = f(x) meets the x-axis
y > mx + c: half-plane above the line; y < mx + c: below
Strict (<, >): dashed boundary; inclusive (≤, ≥): solid boundary
Test point (0, 0): true → shade its side; false → shade the other side
System: solution = overlap (intersection) of all half-planes

Worked examples

1. Is (3, 5) a solution of y = 2x − 1? Is (1, 2)?

For (3, 5): 2 × 3 − 1 = 5 ✔, so it lies on the graph and is a solution. For (1, 2): 2 × 1 − 1 = 1 ≠ 2, so it is not on the graph.

2. Solve 2x − 1 = −x + 2 graphically and check.

Draw y = 2x − 1 (through (0, −1) and (1, 1)) and y = −x + 2 (through (0, 2) and (2, 0)). They cross at (1, 1), so x = 1. Check: 2(1) − 1 = 1 and −1 + 2 = 1 ✔.

3. Using the same graphs, solve 2x − 1 ≤ −x + 2.

We need where the blue line y = 2x − 1 is on or below y = −x + 2. Left of the crossing at x = 1 it is lower, and at x = 1 they are equal. Answer: x ≤ 1.

4. Shade y ≥ −2x + 4. Is (1, 1) a solution?

Boundary y = −2x + 4 is solid (≥) through (0, 4) and (2, 0). Test (0, 0): 0 ≥ 4 is false, so shade the side away from the origin (above the line). For (1, 1): 1 ≥ −2 + 4 = 2? False, so (1, 1) is not a solution.

5. Graph 3x + 2y < 12 and say which side to shade.

Boundary 3x + 2y = 12 meets the axes at (4, 0) and (0, 6); draw it dashed (<). Test (0, 0): 0 < 12 true, so shade the side containing the origin (below the line).

6. Find the corner of the region y > x + 1 and y ≤ −x + 3, and give one point inside it.

Corner: x + 1 = −x + 3 → 2x = 2 → x = 1, y = 2. The corner (1, 2) is on the dashed line, so it is not included. Try (0, 2): 2 > 1 ✔ and 2 ≤ 3 ✔, so (0, 2) is inside.

7. Solve |x| = 0.5x + 3 graphically.

Draw the V-shape y = |x| and the line y = 0.5x + 3. Right branch: x = 0.5x + 3 → x = 6 (point (6, 6)). Left branch: −x = 0.5x + 3 → x = −2 (point (−2, 2)). Two crossings, so x = −2 or x = 6.

Common mistakes

Giving the y-value of the intersection as the answer to f(x) = g(x). The solution is the x-coordinate.
Using a solid line for < or >. Strict inequalities need a dashed boundary because points on the line do not count.
Testing a point that lies on the boundary line. Choose a point clearly on one side, such as (0, 0) if the line does not pass through it.
Shading each inequality of a system but giving the whole shaded area as the answer. Only the overlap satisfies all of them.

Practice quiz

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Solving Equations and Inequalities Graphically

🎬 Step-by-step story

🤔 Common doubts, cleared

Why is a whole line the answer to one equation?

Why do we read only x at the crossing point?

How do I know which side of the crossing is the answer for f > g?

Why dashed for > but solid for ≥?

Why not test any point I like?

What is the answer when two shadings overlap?