Shading the Truth

You already know how to graph a line. But what if the answer isn’t on the line — it’s an entire region of the graph? Welcome to the world of inequalities, where the line is just the boundary and the real action is in the shading.

What Is an Inequality?

An equation like y = 2x + 1 gives you a line. But an inequality like y > 2x + 1 gives you everything above that line. It’s not one answer — it’s infinitely many answers, a whole half of the coordinate plane.

Part 1: Number Line Inequalities

Before we hit the coordinate plane, let’s start simple. On a number line, x > 3 means every number to the right of 3. We draw an open circle at 3 (because 3 itself is NOT included) and shade to the right.

If it’s x >= 3, we use a filled circle because 3 IS included.

Key symbols to remember:

Part 2: The Boundary Line

Now let’s move to two dimensions. Every linear inequality has a boundary line — that’s the line you’d get if you replaced the inequality with an equals sign.

Drag the sliders below to change the slope and y-intercept of the boundary line:

This is just a regular line — the same one from the linear equations lesson. The inequality part comes next: which side do we shade?

Part 3: Shading Above and Below

Here’s the big rule:

• y > mx + b or y >= mx + b — shade above the line
• y < mx + b or y <= mx + b — shade below the line

Let’s see it in action. The boundary line separates two regions. Watch how changing the slope and intercept moves the entire boundary:

The faded line above represents the shaded region. Every point in that region has a y-value greater than the boundary line. Any point you pick above that boundary will satisfy the inequality.

Part 4: Testing a Point

Not sure which side to shade? There’s a foolproof trick: pick a test point and plug it in. The easiest test point is usually (0, 0) — the origin.

For example, does (0, 0) satisfy y > 2x + 1?

Plug in: 0 > 2(0) + 1 becomes 0 > 1. That’s false, so (0, 0) is NOT in the solution region. Shade the other side.

Part 5: Comparing Two Boundary Lines

What if you have two inequalities at once? Now you’re looking for points that satisfy BOTH conditions — the region where the two shadings overlap.

The two lines divide the plane into regions. When you have a system of inequalities, the solution is the region that satisfies ALL of them at once.

Part 6: Compound Inequalities

A compound inequality combines two conditions. For instance:

-3 < y < 3 means y is between -3 and 3.

On a graph, this looks like a horizontal band:

Everything between the two lines is the solution region. Points on the lines themselves depend on whether the inequality is strict or includes equals.

Now let’s do a more interesting compound inequality with sloped boundaries:

The solution is the band between the two parallel lines. Slide the “Band width” to see how the region grows and shrinks.

Wrapping Up

Here’s what you’ve learned about inequalities:

Inequalities aren’t harder than equations — they’re just bigger. Instead of one answer, you get a whole region of answers. And that boundary line? It’s just the fence between “yes” and “no.”