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More than and less than, drawn

Solve inequalities visually

An equation asks where two things are equal. An inequality asks where one is bigger, and the answer is a whole zone, not one point. Pick more than or less than, drag the flat line, and watch the solution light up.

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2x + 1 > 5
  1. 2x + 1 > 5
  2. 2x > 4
  3. x > 2
The blue line is above the flat line when x is more than 2.
Idea 1 · Where is one path above the other?

Where is one path above the other?

To solve 2x + 1 > 5, draw each side as a line: the blue line y = 2x + 1 and the flat line y = 5. The solution is every x where the blue line is higher.

The lines cross at x = 2. To the right of the crossing the blue line is above the flat one, so every x greater than 2 works. That is a whole range, not one number.

Aha!

An equation asks where two paths are equal. An inequality asks where one is above the other: everything on one side of the crossing.

Idea 2 · Shade one side of a line

Shade one side of a line

With two variables, y > x + 1 means every point whose height is above the line y = x + 1. Draw the line, then shade the side where the statement is true.

Check with a test point. (1, 4) gives 4 > 2, which is true, so the side holding (1, 4) is the one to shade. The line itself is dashed, because points on it are not more than.

Aha!

Shade the zone where the statement is true: every point above the line is a solution, and points on a dashed line are not.

Idea 3 · The overlap of two zones

The overlap of two zones

A system of inequalities asks for two things at once, like y > x − 1 and y < 5 − x. Shade each zone on its own first.

A point has to pass both tests, so the solution is where the two zones overlap. Shading the two sides separately, and stopping there, is the usual mistake.

Aha!

A point must pass both tests, so the solution is where the two shaded zones overlap.

Idea 4 · Open and closed circles

Open and closed circles

x > 2 does not include 2, so on the number line it gets an open circle. x ≥ 2 does include 2, so the circle is filled.

The shaded ray shows the whole answer. A strict inequality (> or <) leaves the edge out. “More than or equal to” (≥ or ≤) keeps it in.

Aha!

An open circle leaves the edge out, a filled circle keeps it in, and the arrow shows the answer is a whole range.

Quick answers

Questions about inequalities

How do you solve an inequality with a graph?

Draw each side as a line and find where they cross. The solution is every x where the line for the left side is above (for >) or below (for <) the line for the right side.

What is the difference between > and ≥?

> means strictly more than, so the boundary number is not a solution and gets an open circle. ≥ means more than or equal to, so the boundary is included and the circle is filled.

How do you know which side to shade?

Pick a test point that is not on the boundary line, put it into the inequality, and see whether it is true. If it is, shade the side with that point. If not, shade the other side.

Why does the sign flip when you divide by a negative number?

Multiplying by a negative reverses the order of numbers on the number line. For example 2 < 5, but −2 > −5. So −2x > 6 becomes x < −3.

Now play it

Try it in Pip: Math Adventure

World 3 of Pip: Math Adventure is a short world on inequalities. You solve “more than” and “less than” questions and show the answer as a range or as a region.

It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.

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