Unit 22: Not Equal
Bigger, smaller and in between.
Unit 22 of 28 in Problem-solving maths for kids. Its 7 lessons are Reading the Signs, The Sign That Turns Round, Drawing the Answer, More Than One Step, Inequalities in Real Life, Comparing Sizes and Ranges and Distances — below is everything each one explains, and a question or two from it to try.
The answers are marked in the app by value — 1/2, 0.5 and 2/4 all count — and every one is re-derived by sympy before it ships.
This unit opens with a fortnight’s trial of everything — no card needed — or with a family plan, bought in the iPhone app. The first two units of every track are free for ever. Try it in the app.
🔣 Reading the Signs
Four little signs
a < b means a is less than b. a > b means greater than.
Add a line underneath and it includes being equal: a ≤ b means "less than or the same as".
The wide end always faces the bigger number, which is why some people say the sign is a hungry mouth eating the larger one.
Solving is nearly the same
An inequality is solved exactly like an equation. Do the same thing to both sides until the letter is alone.
x + 4 < 10 becomes x < 6.
The difference is the answer. It is not one number — it is every number below 6.
Try it yourself
Which statement is true?
- -5 < -2
- -5 > -2
- -5 = -2
- -2 < -5
Solve x + 9 < 15. What is the largest whole number x can be?
Answer them in the app
🔃 The Sign That Turns Round
Try it and see
Start with something obviously true: 3 < 5.
Now multiply both sides by -1. You get -3 and -5. But -3 is *bigger* than -5.
So the sign had to turn round: -3 > -5. Multiplying or dividing by a negative flips the inequality. Nobody decided that — the number line did.
Adding never flips it
Only multiplying or dividing by a negative causes trouble.
Adding or subtracting *any* number — positive or negative — leaves the sign exactly as it was, because it slides both sides along the line by the same amount.
So x - 8 < 2 is simply x < 10. No flip.
Try it yourself
You divide both sides of -2x > 8 by -2. What happens?
- It becomes x < -4
- It becomes x > -4
- It becomes x > 4
- Nothing changes
Solve -3x > 12. What is the largest whole number x can be?
Answer them in the app
📏 Drawing the Answer
Open circle or filled in
The answer to an inequality is a stretch of the number line, and there is a convention for drawing it.
An open circle means the end is not included — that is < or >.
A filled circle means it is included — that is ≤ or ≥.
Then an arrow shows which way the answers continue.
Reading it from either end
x > 3 and 3 < x say exactly the same thing. The sign always points at the smaller side, so turning the statement round turns the sign round too.
People write it both ways, and a lot of confusion comes from reading one as the other.
The safe habit: always read towards x, and check which side of the line the arrow would go.
Try it yourself
A number line shows an open circle at 5 with an arrow going right. What does it mean?
- x > 5
- x ≥ 5
- x < 5
- x = 5
Solve x + 3 > 8. What is the smallest whole number x can be?
Answer them in the app
🪜 More Than One Step
Same order as an equation
Solve 3x + 4 < 19 exactly as you would an equation: take the 4 off, then divide by 3.
3x < 15, so x < 5.
The only extra thing to watch is whether you ever divide by a negative. Adding, subtracting and dividing by positives leave the sign completely alone.
With x on both sides
Gather the x terms on whichever side keeps them positive, and you never have to flip the sign at all.
2x + 1 < 5x - 8. Taking 2x from both sides gives 1 < 3x - 8, then 9 < 3x, so 3 < x.
That little choice at the start saves the most error-prone step in the whole unit.
Try it yourself
Solve 3x + 4 < 19. What is the largest whole number x can be?
Solve 5x - 7 ≥ 18. What is the smallest value x can be?
Answer them in the app
🛒 Inequalities in Real Life
The words that mean a sign
Real problems say inequalities in English rather than symbols.
"At least 10" means ≥ 10. "No more than 10" means ≤ 10. "Under 10" means < 10. "Over 10" means > 10.
Getting the sign right from the words is most of the work. The algebra afterwards is lesson 2.
The answer must make sense
Algebra might say x ≤ 6.5 buses. You cannot hire half a bus.
So round towards what the situation allows: at most 6 buses if you are paying for them, but at least 7 if you need to fit everyone in.
The maths gives the boundary. Reading the story tells you which side of it to land on.
Try it yourself
A ride needs you to be "at least 120 cm tall". Which is right?
- height ≥ 120
- height > 120
- height ≤ 120
- height < 120
Pens cost 3 each and you have 20. What is the most you can buy?
Answer them in the app
🔭 Comparing Sizes
Which is bigger, without a calculator
Is 2/3 bigger than 3/5?
Cross-multiply: 2 × 5 = 10 against 3 × 3 = 9. The first is bigger.
That works because both bottoms are positive, so multiplying by them keeps the sign the same way round. With a negative bottom it would flip — which is why this trick comes with a condition.
Squaring can mislead
For positive numbers, if a < b then a^2 < b^2. Comfortable.
But with negatives it breaks: -5 < 2, yet 25 > 4.
Squaring is only safe as a comparison when you already know both sides are positive. That condition is doing real work, and forgetting it is how people prove things that are false.
Try it yourself
Compare 2/3 and 3/5 by cross-multiplying. What is 2 × 5?
And what is 3 × 3?
Answer them in the app
📊 Ranges and Distances
Two conditions at once
2 < x < 9 means both things must hold: x is above 2 and below 9.
To solve 1 < x + 3 < 7, do the same thing to all three parts. Take 3 off each: -2 < x < 4.
It is one inequality that happens to be squeezed from both sides.
Absolute value is a distance
|x| means how far x is from zero, ignoring which side. So |-7| is 7.
That makes |x| < 3 mean "less than 3 away from zero", which is everything between -3 and 3.
And |x - 5| < 2 means "less than 2 away from 5" — everything between 3 and 7.
The mean is never the smallest
Here is an inequality worth carrying around: for any two positive numbers, their average is at least as big as the square root of their product.
(a + b)/2 ≥ sqrt(ab), and they are only equal when a and b are the same.
So of all rectangles with a given perimeter, the square has the largest area. That single fact answers a surprising number of puzzles.
Try it yourself
Solve 4 < 2x < 14. How many whole numbers can x be?
Solve 1 ≤ x - 4 ≤ 6. What is the largest value x can be?
Answer them in the app