Writing Inequalities from Two Points- Step-by-Step Guide

What You're Actually Doing Here

When you have two points on a coordinate plane, you can write an inequality that describes the region containing those points. This isn't some abstract math exercise—it shows up in linear programming, optimization problems, and computer graphics.

The process is straightforward: find the line, then decide which side of that line you want. That's it.

The Basic Formula You Need

For two points (x₁, y₁) and (x₂, y₂), the slope is:

m = (y₂ - y₁) / (x₂ - x₁)

Once you have the slope, plug one point into y - y₁ = m(x - x₁) to get the equation of the line. Then convert it to an inequality by replacing the equals sign with <, >, ≤, or ≥.

Step-by-Step: Writing the Inequality

Let's work with points (2, 3) and (5, 9).

Step 1: Calculate the Slope

m = (9 - 3) / (5 - 2) = 6/3 = 2

Step 2: Write the Line Equation

Using point (2, 3):

y - 3 = 2(x - 2)

y - 3 = 2x - 4

y = 2x - 1

Step 3: Convert to an Inequality

Here's where you need to think. The inequality sign depends on which region you want.

Test a point not on the line. The origin (0, 0) is usually easiest.

Plug it in: 0 ? 2(0) - 1 → 0 ? -1

Since 0 > -1, the region containing (0, 0) satisfies y > 2x - 1

That's your inequality.

How to Determine the Correct Sign

Most students get stuck here. Here's the rule:

• Pick any point not on the line
• Substitute the coordinates into your equation
• If the result is true, that point satisfies the inequality—and so does every point on the same side of the line
• If false, you need the opposite sign

That's literally all there is to it. The sign isn't arbitrary—it tells you which half-plane you want.

Handling Vertical Lines

What if your two points have the same x-coordinate? Like (3, 1) and (3, 7)?

You can't use slope—you don't have one. The line is x = 3.

For the inequality:

• x < 3 means everything to the left
• x > 3 means everything to the right

Test with any point that has a different x-value. (0, 4) gives x = 0, which is less than 3, so it satisfies x < 3.

Working with Bounded Regions

Sometimes you need an inequality that passes through two points and satisfies a condition. Example:

Write an inequality through (1, 2) and (4, 8) where the region is below the line.

Line: y = 2x

"Below" means y < 2x. Test (0, 0): 0 < 0 is false, so (0, 0) is not in the region. That's correct—"below" means smaller y-values, and the origin has y = 0 while the line at x = 0 has y = 0. They're equal, not less.

Try (0, -1): -1 < 0 is true. That point is below the line.

Quick Reference: Inequality Signs and Meanings

The solid vs. dashed line distinction matters when you're graphing. If your inequality includes "or equal to," draw a solid line. Otherwise, use dashes.

Common Mistakes That Waste Time

• Forgetting to test a point. The sign you choose must be verified, not guessed.
• Getting the inequality backwards. "Above" the line means larger y-values, so y > mx + b.
• Losing track of which point you used. Use either point in the point-slope formula—you'll get the same line either way.
• Rushing the slope calculation. A sign error here breaks everything downstream.

Practice: Find the Inequality

Points: (1, 4) and (3, 10). Write the inequality for the region containing (0, 0).

Slope: (10 - 4) / (3 - 1) = 6/2 = 3

Line: y - 4 = 3(x - 1) → y = 3x + 1

Test (0, 0): 0 ? 3(0) + 1 → 0 ? 1

0 < 1 is true. Answer: y < 3x + 1

When You'll Actually Use This

Linear programming problems ask you to maximize or minimize something subject to constraints. Those constraints are inequalities like the ones you just learned to write. Each constraint comes from a boundary line determined by two points.

Computer graphics uses half-plane inequalities to determine which side of a boundary a pixel falls on. Same math, different application.

The skill transfers. Learn it properly now and you won't relearn it later.