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📐 Mathematics - Class 7
Lesson 24 · Equations, Inequalities and Line Graphs

Inequalities on the Number Line

MathematicsSubject
13 minEstimated read
Almost an equation, with one exception

The same moves as an equation, with one exception: multiplying or dividing by a negative flips the sign round.

Symbol Reads as On the number line
x > 3x is greater than 3Open circle at 3, arrow right
x < 3x is less than 3Open circle at 3, arrow left
x ≥ 3x is greater than or equal to 3Filled circle at 3, arrow right
x ≤ 3x is less than or equal to 3Filled circle at 3, arrow left
An inequality has many answers, not one

3x + 5 = 20 gives one value. 3x + 5 > 20 gives every value above 5, which is infinitely many.

The open and filled circle

An open circle means the number itself is not included; a filled one means it is. That small mark carries real information.

Multiplying or dividing by a negative flips the sign

3 < 5 is true. Multiply both sides by −1 and you get −3 and −5, but −3 is GREATER than −5, so the sign must turn round.

Adding a negative does not flip it

Subtracting 5 from both sides leaves the sign exactly as it was. Sliding both numbers along the line does not change which is further right.

Avoiding the flip altogether

Move the negative x term instead of dividing by a negative. 10 − 2x > 4 becomes 10 > 4 + 2x, giving x < 3 with no flip needed.

Check with a number from your answer

Try a value your answer allows, then a value it excludes. Both tests together confirm the direction of the sign.

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Worked Example 1

Solve 3x + 5 > 20 and show the answer on a number line.

Solve it exactly like an equation3x > 15, then x > 5
Draw itOpen circle at 5, arrow to the right
Check with two numbers6 works (23 > 20) and 4 correctly fails (17 is not > 20)
Worked Example 2

Solve −2x > 6.

The x term is negativeDividing by −2 is the move that flips the sign
Divide and flipx < −3
Why the flip is necessary3 < 5 becomes −3 > −5. The order of the whole line reverses.
Check itx = −4 works, and x = 0 correctly fails.
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Activity 24.2

The number line walk

What you needA number line chalked across the floor from minus 10 to 10, and cards with values on them.
MethodCall out an inequality. Everyone whose card satisfies it stands on their number.
Then test the flipCall out the same inequality multiplied by minus 1 and see who swaps.
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1

An inequality says one side is larger than the other, rather than equal to it.

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1How is solving an inequality different from solving an equation?
2Why does multiplying by a negative number flip the inequality sign?
3What is the difference between an open circle and a filled circle on a number line?
4How can you solve an inequality with a negative x term without flipping the sign?
5How should you check the answer to an inequality?

Question 1 of 10

1Solving an inequality uses
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Slide 1 of 4
Unit 24 · Inequalities

Inequalities

Mathematics · Grade 7

What You Will Be Able To Do

↔️

Read and write the four inequality symbols correctly

📏

Show an answer on a number line, with the right kind of circle

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Know exactly when the sign flips, and prove why

✅

Check an answer with two numbers, one allowed and one not

“

Solving an inequality uses exactly the same moves as solving an equation, with a single exception: multiplying or dividing by a negative number flips the sign round.

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Presenter notes: Open with the question left hanging at the end of last lesson. Write 3 is less than 5 on the board and ask whether it is true, which everybody agrees it is. Then multiply both sides by minus one in front of them and ask whether the new statement is still true. It is not, and that single demonstration is the reason this lesson exists.