Write The Inequality For The Graph Below

Reading the Inequality Straight From a Graph (And Why Most People Get It Backwards)

You've got a graph in front of you — a line, maybe a few shaded dots, possibly a dashed line instead of a solid one — and the question is simple on the surface: what's the inequality?* But anyone who's stared at one of these for more than thirty seconds knows the simple part ends the second you have to decide whether the line is solid or broken, or whether the shading is above or below.

Here's the thing — once you know what to look for, these graphs aren't hard. They're just easy to misread if you rush. So let's slow down and actually work through what every part of the graph is telling you.

What the Graph Is Actually Showing

An inequality graph is a picture of every single point that makes an inequality true. Not just one solution. Not two. All of them.

When you graph an equation like y = 2x + 1*, you get a single line, and only the points on that line count. But the moment you swap that equals sign for a less-than, greater-than, less-than-or-equal-to, or greater-than-or-equal-to, the line either opens up to include everything above it, everything below it, or — in the strict cases — everything except the line itself.

So really, the graph is doing two jobs at once: it's showing you the boundary line, and it's showing you which side of that line is "winning."

The Boundary Line: Your Starting Point

The line on the graph is just the related equation with the inequality sign replaced by an equals sign. If you can find two points on that line, or its slope and y-intercept, you can write the equation. From there, the inequality is one symbol away.

Here's one way to look at it: a line passing through (0, 3) with a slope of 2 has the equation y = 2x + 3*. That part is straightforward. The trick is figuring out the symbol*.

The Shading: Your Big Clue

The shaded region of the graph is the set of points that satisfy the inequality. Everything inside the shading works. Everything outside doesn't.

So the question becomes: which side is shaded, and what does that side mean mathematically?

• If shading is above the line, the inequality usually looks like y > ...* or y ≥ ...*
• If shading is below the line, it usually looks like y < ...* or y ≤ ...*

That "usually" is doing some work, though. It depends on whether you solved for y first.

How to Write the Inequality Step by Step

The process is the same every time, even when the graph looks different. Here's how I'd walk through it.

Step 1: Identify Two Points on the Line

Pick any two clear points the line passes through. Whole-number coordinates make life easier, but any pair works. Let's say the line crosses (0, -2) and (3, 4).

Step 2: Find the Slope

Slope is rise over run. From (0, -2) to (3, 4), you go up 6 and right 3, so the slope is 2. That gives you m = 2* in the slope-intercept form.

Step 3: Find the Y-Intercept

The y-intercept is where the line crosses the y-axis. Consider this: from the first point, it's already -2. So the equation of the line is y = 2x − 2*.

Step 4: Look at the Line Type

This is the step most people blow past. Is the line solid or dashed?

• A solid line means the points on the line are included. Use ≤ or ≥.
• A dashed line means the points on the line are not included. Use < or >.

This matters because y ≤ 2x − 2* and y < 2x − 2* describe nearly the same region — but the difference between them is a single line's worth of points.

Step 5: Test a Point in the Shaded Region

Pick any point clearly inside the shaded area — (0, 0) is usually a safe bet unless it's on the line. Plug it into your equation.

For y = 2x − 2*, plugging in (0, 0) gives 0 = -2, which is false. So (0, 0) does not satisfy the equation y = 2x − 2*. If (0, 0) is in the shaded region, then we need an inequality that makes 0 = 2(0) - 2 false. That means we want y to be less than* 2x - 2 in that area.

If the shading is below the line and the line is dashed, your final answer is:

y < 2x − 2

If the line is solid, it's y ≤ 2x − 2*. That's it. No magic.

Common Mistakes That Mess People Up

Mixing Up Solid and Dashed

A dashed line isn't a stylistic choice. Now, it tells you something. y < 2x − 2* and y ≤ 2x − 2* look almost identical on a graph, but only one of them is correct based on whether that line is drawn solid or broken.

Forgetting to Flip the Sign

When you multiply or divide both sides of an inequality by a negative number, you have to flip the sign. Say you start with −2x + y ≤ 4 and you want to get y alone. But if you started with 2x − y < 6, you'd have to divide by −1 to isolate y, and suddenly it becomes y > −2x − 6*. Because of that, this trips people up when moving from standard form to slope-intercept form. No flip needed. You'd add 2x to both sides and get y ≤ 2x + 4*. That sign flip is the kind of thing that quietly wrecks an answer.

Want to learn more? We recommend how many days in 10 weeks and what is 14 days from today's date for further reading.

Assuming Above Always Means Greater Than

It does — but only if y is isolated on the left side of the inequality. If the inequality is written as 2x + y > 4, "above the line" still represents a true statement, but the symbol on the page is greater-than, not less-than. Always check which variable is by itself.

Shading Without a Test Point

Eyeballing which side should be shaded works about 80% of the time, but not always. That's why pick a point. If the slope is negative, your intuition can flip. Plug it in. Trust the math, not your gut.

Confusing Vertical and Horizontal Lines

A vertical line at x = 3* is not the same kind of problem. The inequality just becomes x > 3*, x < 3*, x ≥ 3*, or x ≤ 3* — and the shading runs left or right, not up or down. There's no y in the boundary equation. Same rules apply, but the geometry looks totally different.

Practical Tips That Actually Help

Start with the easy stuff. Find the y-intercept first. If it's a whole number, you're already halfway to writing the equation.

Use intercepts whenever you can. If the line crosses both axes cleanly, the x-intercept and y-intercept give you two points without any guesswork.

Always do a sanity check. Pick a point in the shaded area. Pick a point outside it. Plug both into your final inequality. The shaded one should make the inequality true. The unshaded one should make it false. If both work, something's off.

Rewrite in slope-intercept form before you finalize. Even if the graph is in standard form, converting to y = mx + b* makes the meaning of the shading obvious.

Don't forget the line type at the end. Write the inequality, then* check whether the line is solid or dashed. It's surprisingly easy to skip that final glance.

FAQ

How do I know if it's ≤ or <?

Look at the line. Solid = the points on the line count, so use ≤ or ≥. Worth adding: dashed = the points on the line don't count, so use < or >. No exceptions.

What if the line is vertical?

The boundary equation is just x = some number*, and your inequality is going to be x <, x >, x ≤, or *

x ≥* that same number. Shade left for less-than, right for greater-than.

What if the line is horizontal?

A horizontal line at y = 5* becomes y = 5*, and the inequality is straightforward — y > 5* shades above, y < 5* shades below. No special handling required; it behaves like any other line with a slope of zero.

Can I graph without converting to slope-intercept form?

Technically, yes. You can plot the line using two points found from standard form, then test a point to decide shading. But slope-intercept form makes the shading direction predictable, and it removes one common source of errors.

What if the inequality has x and y on the same side?

That's standard form, like 3x + 2y ≥ 12. Graph the line using intercepts, isolate y to make shading clearer, or just use a test point. Both methods work.

Why does dividing by a negative number flip the inequality?

Because multiplying or dividing both sides of an inequality by a negative number reverses the order — the same reason a number line flips when you multiply by −1. It's not arbitrary; it's a property of how negative values relate to each other.

Common Test Scenarios to Watch For

Two-variable systems with a twist. Sometimes you'll be given the graph and asked to write the inequality. Match the boundary line first, then pick a shaded point and plug it in to determine the symbol.

Mixed equality types. A problem might give you y > 2x − 3* with a dashed line but then ask for the region that includes the origin. Make sure the test point you choose actually falls in the shaded region.

Slope of zero or undefined. These look weird at first but follow the same rules. A zero-slope line is horizontal; an undefined-slope line is vertical. Once you identify which one you're dealing with, everything else is the same process.

Fractions in the equation. Multiply everything by the denominator first to clear fractions. It's a small step, but it prevents arithmetic errors that throw off the entire graph.

A Quick Recap

The whole process boils down to four moves: identify the boundary, determine whether it's solid or dashed, isolate y when possible, and test a point to confirm the shading. The rest is just paying attention to small details — sign flips, the line type, the variable that's isolated.

Graphing linear inequalities isn't glamorous, but it's one of those skills that pays off in every later math class. Statistics uses it. On top of that, even physics leans on it more than you'd expect. Calculus uses it. So getting comfortable now saves a lot of confusion later.

And if you ever get stuck, remember: the math is always right. If your test point disagrees with your shading, trust the point. Adjust the graph. Move on.