• A 2D inequality (e.g. y < x or x + y ≥ 8) involves two variables, and its solution is a whole region of the xy-plane, not just a line.
• To draw a region: turn the inequality into an equation, draw that as a straight line, then decide which side is "wanted."
• The line is solid for ≤ or ≥ (the line itself counts), and dotted for < or > (the line itself doesn't count).
• Convention: we usually shade the unwanted side of each line, so the clean, unshaded patch left over is your answer region — normally labelled R.
• Vertical lines (x = k) and horizontal lines (y = k) follow the same left/right and above/below logic.
• You can also work backwards: given a shaded diagram, read off the equations of the lines and figure out which inequality signs produced that shading.
• If you're ever unsure which side is which, test a point — plug its coordinates into the inequality and see if it holds true.

From lines to regions

You already know that an equation like y = 2x + 3 draws a single, precise line on a graph — every point on that line satisfies the equation exactly, and no other point does.

Now swap the equals sign for an inequality sign: y > 2x + 3. Suddenly you're no longer asking "which points sit exactly on this line?" You're asking "which points have a y-value bigger than the line gives you at that x?" That's not one line anymore — it's an entire half of the plane, infinite in size, sitting above the line.

That's the whole shift in thinking this chapter asks of you: an equation is a boundary; an inequality is everything on one side of that boundary. The line itself just tells you where the boundary sits — the inequality sign tells you which side you actually want.

Step 1: turn the inequality into an equation

Before you can shade anything, you need the boundary itself. Take whatever inequality you're given and simply replace the inequality sign with an equals sign. So 3x + 2y ≥ 12 becomes 3x + 2y = 12. Now you draw that as an ordinary straight line — rearranging into y = mx + c form first if it helps you plot it.

For example, rearranging 3x + 2y = 12:

2y = -3x + 12

 y = -3/2 x + 6      ← gradient -1.5, y-intercept 6

Step 2: solid or dotted?

This is the detail examiners love to check, because it's easy to forget under exam pressure. Ask yourself: does the boundary line itself count as part of the answer?

The quick rule for sloped and horizontal lines

Once your line is written in the form y = ..., the inequality sign tells you the side directly:

This makes sense if you think about it logically: if y has to be less than the line's value at every x, then you're looking underneath the line. If y has to be greater than, you're looking above it. The direction of the inequality sign literally points you up or down.

Vertical lines are a special case

Vertical lines don't have a "y = ..." form (their gradient is undefined), so they use left/right instead of above/below:

When in doubt: the point-test method (your safety net)

If you ever second-guess yourself — and everyone does under exam stress — there's a foolproof backup: pick any point that is clearly not on the line (the origin (0,0) is usually easiest, as long as the line doesn't pass through it), substitute its coordinates into the original inequality, and see whether the statement comes out true or false.

• If it's true, that point — and the whole side of the line it sits on — is in the wanted region.
• If it's false, that side is the unwanted region, and the wanted side is the other one.

Shade what you don't want

Here's the part that trips students up at first, because it feels backwards: the standard convention (unless a question explicitly says otherwise) is to shade the unwanted region of each line — not the wanted one.

Why? Because when you have three or more inequalities combined together, the true answer region is wherever all the conditions overlap. If you shaded the wanted regions instead, you'd end up hunting for the one spot buried under multiple layers of shading, which is messy and easy to misread. Shade the unwanted parts instead, and whatever is left completely white/unshaded is automatically your answer — clean and unambiguous.

        y

        │        ░░░░░  ← shaded (unwanted: above y = 2x)

        │      ░░░░░░░

        │    ░░R░░░░░  ← R = clear, unshaded triangle

        │  ░░░░░░░░

        │░░░░░░░░

        └──────────────── x

             ░ = shaded away (unwanted)

Given a shaded region, find the inequalities

Exam papers often flip the question around: instead of asking you to draw the region, they show you a finished graph with a shaded (or unshaded) area and ask you to state the inequalities that produced it. This is really just the whole process in reverse, done systematically:

1. Find the equation of each line. For sloped lines, read the gradient and y-intercept to build y = mx + c. For vertical lines, it's simply x = k; for horizontal lines, y = k.
2. Check solid vs dotted for each line — this tells you whether to use ≤/≥ (solid) or </> (dotted).
3. Look at which side is shaded/wanted relative to each line, and apply the above/below or left/right rule to pick the correct inequality sign.
4. Double-check with a point from inside the marked region if you're not confident — substitute its coordinates in and confirm the inequality holds true.

Fully worked example

Suppose a graph shows a shaded triangle bounded by three lines: a solid vertical line at x = 1, a dotted line through the origin, and a solid line with y-intercept 7 and gradient -1. A point marked inside the region is (2, 4).

  y

  8│  x=1        y=x (dotted)

  7│──●              ╱

  6│  │            ╱

  5│  │          ╱

  4│  │   ● (2,4)

  3│  │      ╲

  2│  │        ╲

  1│  │          ╲  y = -x+7 (solid)

  0└──┴────────────── x

     1  2  3  4  5  6  7

Line 1 — x = 1 (solid): the shaded region sits to the right, so: x ≥ 1

Line 2 — y = x (dotted): testing (2,4): "4 > 2" is true, so the region is above the line: y > x

Line 3 — y = -x + 7 (solid): testing (2,4): "4 ≤ -2+7" → "4 ≤ 5" is true, so the region is below/on the line: y ≤ -x + 7