Which Inequality Is Shown In The Graph Below

Reading a Graph Like a Detective

Ever stare at a math graph and feel like it's speaking a language you didn't sign up for? Day to day, you're not alone. Most students hit a wall when a problem says "which inequality is shown in the graph below" because the question assumes you can translate a picture into algebra. The thing is, it's not as hard as it looks once you know what to look for.

So let's break this down properly. By the end, you'll be able to look at a graph and write out the matching inequality without second-guessing yourself.

What the Question Is Actually Asking

When a problem says "which inequality is shown in the graph below," it's asking you to reverse-engineer an inequality from a visual. The graph shows a region on a coordinate plane, and that region represents every single point that satisfies some inequality. Your job is to figure out the rule that produces that region.

The main things you need to identify are:

The Boundary Line

The boundary is the line that separates the shaded region from the unshaded region. Two things matter here:

• Solid or dashed? A solid line means the points on the line are included* in the solution — so the inequality uses ≤ or ≥. A dashed line means points on the line are excluded* — so the inequality uses < or >. This one detail catches a lot of people.
• What's the equation of the line? You read the slope and y-intercept directly from the graph, just like you would for a regular linear equation. If the line passes through (0, 2) and rises 1 for every 2 across, the equation is y = (1/2)x + 2. Same rules as always.

The Shaded Region

The shaded area tells you which side of the line satisfies the inequality. In practice, pick any point clearly inside the shaded region and plug it into the inequality. And if it makes the statement true, you've got the right direction. If not, flip the sign.

A quick trick: test the origin (0, 0) if it's not on the boundary line. Plugging in zero is easy, and it tells you almost immediately which side of the line you're on.

Why This Type of Problem Trips People Up

The reason this question causes so much confusion isn't the math — it's the visual translation. That said, you're doing two things at once: reading a graph and writing algebra. That switch between modes is where mistakes creep in.

And there's another layer. Most textbooks show you how to graph* an inequality, then assume you can do the reverse. They rarely slow down and teach the reverse process as its own skill. So you end up practicing one direction and getting tested on the other. Not a great setup.

In practice, the most common errors are:

• Writing = instead of an inequality symbol, because you focused on the line and forgot the shaded region exists.
• Mixing up < and > because you didn't bother to test a point.
• Assuming every line is a function. Some boundary lines are vertical (x = something), and a vertical line on a graph means the answer involves x, not y. That alone flips your whole approach.

How to Solve It Step by Step

Here's the method that actually works. Use it every single time, and the problem becomes almost mechanical.

Step 1: Identify the Type of Boundary Line

Look at the line. In practice, is it solid or dashed? Here's the thing — write down ≤ or ≥ if it's solid, and < or > if it's dashed. Don't worry about which one yet — just lock in whether equality is included.

Step 2: Find the Equation of the Line

Read two clear points off the graph. Write the line in slope-intercept form: y = mx + b. Find the y-intercept (or the x-intercept, depending on whether the line is vertical or horizontal). Calculate the slope. If the line is vertical, write it as x = some number instead.

Step 3: Replace the Equals Sign with an Inequality Sign

Now swap the = in your equation for the correct inequality symbol. The line equation becomes the framework — only the sign still needs to be determined.

Step 4: Test a Point in the Shaded Region

Pick a point that's clearly inside the shaded area. And plug the x and y values into your inequality-in-progress. That's why if the statement is true, you're done. If it's false, flip the inequality sign.

That's it. Four steps. No magic.

Worked Example to Make It Stick

Say the graph shows a solid line passing through (0, 3) and (2, 1), with the region above the line shaded.

The line is solid, so the inequality is either y ≥ mx + b or y ≤ mx + b. Worth adding: the slope between those two points is (1 - 3) / (2 - 0) = -1. The y-intercept is 3. So the line is y = -x + 3, and the inequality is either y ≥ -x + 3 or y ≤ -x + 3.

For more on this topic, read our article on which expression has a value of 10 or check out how many days are in 11 months.

Now test a point in the shaded region — say (0, 5). Plug it in: is 5 ≥ -(0) + 3? Yes, 5 ≥ 3 is true. So the answer is y ≥ -x + 3.

Simple. The whole thing took about thirty seconds once you know the pattern.

Common Mistakes and How to Dodge Them

Even after you get the hang of it, a few traps will keep showing up. Here's what to watch for.

Ignoring Whether the Line Is Dashed

This is the single most common mistake. Students see the line, write down the equation, and forget to check whether the boundary is solid or dashed. Then they write ≤ when they should have written <, and the whole answer is technically wrong.

Quick fix: before writing anything, ask yourself — "Is this line solid or dashed?" Out loud, if you have to. Make it a habit.

Mixing Up Above and Below

Which side of the line is shaded? Worth adding: don't try to eyeball it. Your eyes will lie to you, especially on a tricky graph. If you get confused, just test a point. Pick a clear point in the shaded region, plug it in, and let the math decide.

Forgetting Vertical and Horizontal Lines

A vertical line at x = -2 with the region to the right shaded gives you x > -2, not y > something. People default to y = mx + b form every time, and that breaks the moment the line isn't diagonal.

If the line is vertical, the inequality only involves x. Which means if it's horizontal, the inequality only involves y. That's a freebie — you don't even need a slope.

Practical Tips That Actually Help

• Always start with the line type. Solid or dashed. Get that written down before anything else.
• Keep slope-intercept form in your back pocket. Even if the line is in some other form on the graph, rewriting it as y = mx + b makes the inequality step cleaner.
• Test (0, 0) when you can. If the origin is in the shaded region, plugging it in gives you an instant true/false check. If the origin is on the line, pick a different point — the test only works when the point isn't on the boundary.
• Don't overthink it. This isn't a trick question. It's a four-step process. The more mechanical you make it, the fewer mistakes you'll make.

FAQ

Does a solid line always mean ≤ or ≥?

Yes. A solid line means the points on the line count as part of the solution, so the inequality must include equality. Dashed always means strict inequality (< or >).

What if the graph shows a parabola or a curve instead of a line?

Then the inequality involves a quadratic, not a linear equation. The process is the same — identify the curve, write its equation, check whether it's solid or dashed, then test a shaded point to determine the direction. The only difference is the equation itself will have an x² term.

How do I know whether to shade above or below the line?

Test a point. Don't guess. Worth adding: pick a point in the shaded region, plug it in, and let the math tell you whether the inequality is greater than or less than. It takes five seconds and removes all guesswork.

What if the shaded region is on a vertical line itself?

That just means x equals a specific number with a strict or non-strict inequality. As an example, a solid vertical line at x = 4 with no shading on either side isn't really an inequality region — it's just the line. If there's a small shaded strip along a vertical line, the inequality is something like

...x > 4 or x < 4 depending on which side is shaded. The vertical line itself is the boundary, and the shading indicates the direction of the inequality.

What about nonlinear boundaries like circles or absolute value graphs?

The same four-step process applies. Identify the equation of the boundary curve (e.Test a point from the shaded region to decide the inequality direction. Which means g. , x² + y² = 9 for a circle). Determine if it's solid (≤ or ≥) or dashed (< or >). For a circle, shading inside means x² + y² < 9 (if dashed), while shading outside means x² + y² > 9.

The Bottom Line

Graphing inequalities is a mechanical skill, not a test of intuition. By breaking the problem down into a consistent sequence—identifying the boundary, determining the inequality type, choosing a test point, and verifying the direction—you remove the guesswork. The graph is just a visual representation of a logical rule. Follow the steps, trust the math over your eyes, and the correct inequality will emerge every time. It's less about "seeing" the answer and more about systematically deducing it.