An equation pins the unknown to a single point. An inequality opens it up to a whole stretch of the line. Stage 12 lives on that picture.
Equals is a wonderful word, but it only tells you when two things match
exactly. Real life is mostly not exact. You are a little taller than your
deskmate. The bridge ahead is rated for no more than 5 tons. You must be at least
12 to ride the big slide. None of those are "equals" — they are "bigger," "smaller," "at most,"
"at least." Math has its own short, sharp symbols for these, and once you can read them you can
describe the whole messy, unequal world.
By the end of this lesson you will read the four inequality symbols
><≥≤ aloud without hesitating, turn an everyday sentence into one of them,
picture what it says on the number line, and settle "which number is bigger?" by a single clean
trick — subtraction. Throughout Stage 12 we keep a steady set of colors:
blue for the number line and the variable, amber
for the bigger side or a boundary value, green for "true here" (the
solution), and red for a trap or a value that is not allowed.
Watch for them and the pictures will read themselves.
12.1.1 Unequal relationships all around us
Pick any two things you can measure and they are almost never equal. One backpack is heavier.
One runner is faster. One price is cheaper. Whenever we compare two amounts, four little words
cover almost every case: bigger, smaller, at least, at most. Each one
has a symbol, and each symbol has a name we read out loud.
The two strict symbols compare without any "or equal":
> means greater than and < means
less than. So 7 > 4 is read
"seven is greater than four," and 4 < 7 is read
"four is less than seven." Same fact, said from each side.
The two inclusive symbols add the words "or equal to":
≥ is greater than or equal to and ≤ is
less than or equal to. These are exactly the symbols we reach for when everyday speech says
at least / no less than (that is ≥) or at most /
no more than (that is ≤). The little line under the symbol is the
"or equal" promise: the boundary value itself is allowed in.
Four symbols, two pairs. The bar underneath turns > into ≥ and < into ≤ — the "or equal" promise.
Here is the one reading trick worth memorizing. The symbol has a small, pinched end and a
wide, open end. The small end always points at the smaller number;
the wide end opens toward the bigger one. So in 4 < 7
the point aims at the 4 (the small one); in 7 > 4
it aims at the 4 again. The symbol is like a hungry mouth that always wants to gobble the larger
amount.
Key idea
An inequality is a sentence about order — which side is bigger.
> "greater than", < "less than",
≥ "at least / no less than", ≤ "at most / no
more than." The pinched point aims at the smaller number.
Everyday
"The elevator holds no more than 8 people" → at most 8 →
p ≤ 8 (8 is fine).
"You must be at least 12 to ride" → no less than 12 →
age ≥ 12 (exactly 12 rides).
🎮 Try it WORDS → SYMBOL TRANSLATOR
Tap a real-life phrase. See which symbol it becomes and whether the boundary value is counted in.
12.1.2 Turning words into symbols
Reading a symbol is half the job; the more useful half is going the other way — taking a
sentence and writing the inequality it hides. Two questions do the whole translation:
which way (greater or less?) and is the boundary itself allowed in (strict or
inclusive?). Get those two right and you are done.
Start with a name for the unknown amount, then translate the key phrase:
The sentence
The inequality
Boundary counted?
"At least 18 years old"
x ≥ 18
yes — 18 counts
"Speed must not exceed 120"
v ≤ 120
yes — 120 is OK
"Fewer than 30 students"
n < 30
no — 30 is too many
"No more than 5 tons"
w ≤ 5
yes — exactly 5 is allowed
Notice how the deciding word lives in the phrasing. "At least" and "no less than"
let the boundary in, so they are ≥. "At most," "no more than,"
and "must not exceed" also let the boundary in, so they are ≤.
But "fewer than," "more than," "over," and "under" leave the boundary
out — those are the strict ones, < and
>.
Watch out
"At least 18" does not mean x > 18
— that would shut out an 18-year-old who is clearly allowed. It means
x ≥ 18. The single most common slip in this
whole lesson is dropping the "or equal" line. Always ask: does the exact boundary
value still count? If yes, use ≥ or ≤.
The dot tells the whole story. Inclusive “at least 18” gets a
filled dot — the boundary 18 is in. Strict “fewer than 30”
gets an open dot — the boundary 30 is left out.
🎮 Try it STRICT OR INCLUSIVE?
Set a boundary value, then choose how the sentence is worded. Watch the symbol switch between strict and inclusive, and see whether the boundary itself is in the solution.
Boundary value12
Wording
12.1.3 Seeing size on the number line
Symbols are tidy, but the picture is what makes them obvious. Lay the numbers out along a line,
the way a ruler does, with smaller to the left and bigger to the right. Now order has a
shape you can point at. The whole of inequalities lives on this picture, so it pays to trust it
completely.
Here is the rule in one sentence, and it never lets you down:
Key idea
a < b simply means a sits to the
LEFT of b on the number line. Bigger is always further right.
To compare any two numbers, just ask: which one is further left?
For example, 2 < 5 because 2 is to the left
of 5 — no surprise. The picture really earns its keep with negative numbers, where our
instincts can lie to us.
On the line, −5 < −2: −5 sits
further left, so it is the smaller number — even though "5" feels bigger than
"2." Left is small; right is big. Always.
Classic trap
Is −5 bigger or smaller than −2? The digit 5
is bigger than the digit 2, so it feels like −5 should be the larger. It is not.
On the line −5 is further left, so −5 < −2.
Picture a thermometer: −5° is colder, i.e. lower, i.e. smaller. The further left you go, the
smaller the number — even when the digits look big.
🎮 Try it WHICH IS BIGGER? ON THE LINE
Slide a and b along the line. The readout tells you the relation and the reason — which point is further left. Try the negative preset.
a3
b5
12.1.4 Comparing two numbers by subtraction
The number line is perfect for friendly whole numbers. But how do you compare
57
and 710?
You cannot eyeball those. Here is a method that always works, for any two numbers, no
matter how awkward: subtract one from the other and read the sign of the answer.
The difference method
To compare a and b, look at
a − b:
a − b > 0⇒a > b ·
a − b = 0
⇒ a = b ·
a − b < 0⇒a < b
Why it works: if a is bigger, you have something left over after
taking b away, so the difference is positive. If a is smaller, you've
overshot, so the difference is negative. Subtraction turns "which is
bigger?" into the easy question "is this positive or
negative?"
Now the fractions. Put them over a common denominator so the subtraction is clean.
The denominators are 7 and 10, so use 70:
The difference is +1/70, a positive number, so
5/7 > 7/10. (As decimals: 5/7 ≈ 0.714 and
7/10 = 0.700, which agrees.)
The same trick works on whole expressions, not just numbers — and that is where it
becomes powerful. Take any number a and compare
a² + 1 with 2a. Subtract:
a² + 1 − 2a =(a − 1)²≥ 0
A square is never negative — it is zero only when the thing inside is zero. So
(a − 1)² is positive for every a except
a = 1, where it is exactly 0. That means
a² + 1 − 2a is never negative, which says
A first taste of a "famous" inequality
a² + 1 ≥ 2a is true for
every number a, with the two sides equal only at a = 1.
Test it: at a = 3, the left side is 10 and the right is 6 (10 ≥ 6 ✓);
at a = 1, both are 2 (equal ✓). You just proved an inequality that holds
for infinitely many numbers — using nothing but subtraction and "a square is never negative."
We meet ideas like this again in 12.6.
🎮 Try it THE DIFFERENCE-METHOD MACHINE
Pick a pair to compare. The machine computes a − b, colors its sign, and reads off the verdict. For the algebra pair, slide a and watch (a−1)² stay ≥ 0.
a3
★ The big ideas, in one breath
An inequality is a sentence about order. > and
< are strict (boundary left out); ≥ and
≤ are inclusive (boundary counts) — these are the symbols for
"at least" and "at most." The pinched end always points at the smaller
number. On the number line, a < b
just means a is to the left of b, so further left is always smaller —
−5 < −2. And when the numbers are too awkward to
eyeball, the difference method settles it: subtract and read the sign —
positive means the first is bigger, negative
means it is smaller. That one trick even proves a² + 1 ≥ 2a
for every number.
What's next
So far an inequality has been a statement we check. Next, in
12.2, we start
working with them — what you may add, subtract, multiply, and divide on both sides
while keeping it true. There is one move that flips the whole thing around (multiplying by a
negative), and it becomes the star of the stage. From there:
12.3 solving,
12.4 systems,
12.5 quadratics,
12.6 the basic
inequality, and 12.7
real-world uses.
✎ Exercises 12.1
Read each aloud, then say which number is bigger:
(a) 9 > 4
(b) −1 < 6
(c) 0 > −3.
Show answer
(a) "nine is greater than four" — 9 is bigger. (b) "negative one is less
than six" — 6 is bigger. (c) "zero is greater than negative three" — 0 is bigger
(it sits to the right of −3).
Translate into an inequality, using x for the unknown:
(a) "at least 18"; (b) "no more than 5 tons (use w)"; (c) "fewer than 30 students (use n)";
(d) "speed must not exceed 120 (use v)".
Show answer
(a) x ≥ 18
(b) w ≤ 5
(c) n < 30
(d) v ≤ 120. The "at least" and the two
"no more / must not exceed" ones include the boundary (≥, ≤); "fewer than" leaves it out
(<).
A sign reads "You must be over 1.4 m tall to ride." Write it (use h), and
say whether a child exactly 1.4 m may ride.
Show answer
h > 1.4. "Over" is
strict, so 1.4 m is not enough — a child exactly 1.4 m may not ride.
(If the sign said "at least 1.4 m," it would be h ≥ 1.4
and 1.4 m would just make it.)
Put each set of numbers in order from smallest to largest:
(a) 3, −4, 0, −1; (b) −2, −7, −5.
Show answer
(a) −4 < −1 < 0 < 3.
(b) −7 < −5 < −2.
Read left to right on the line: the further left, the smaller — so among negatives the one
with the biggest digits is actually the smallest.
Fill in > or <:
(a) −5 □ −2; (b) −8 □ −10;
(c) −1 □ 0.
Show answer
(a) −5 < −2 (−5 is further
left). (b) −8 > −10 (−8 is to the right of
−10). (c) −1 < 0.
Use the difference method to compare
57
and 710.
Show the subtraction.
Show answer
5/7 − 7/10 = 50/70 − 49/70 = 1/70.
Since 1/70 > 0, we get
5/7 > 7/10.
Use subtraction to compare 3 + 2 and 4.
Show answer
(3 + 2) − 4 = 5 − 4 = 1.
Since 1 > 0,
3 + 2 > 4.
Show that a² + 1 ≥ 2a for every
number a, and say exactly when the two sides are equal.
Show answer
a² + 1 − 2a = (a − 1)².
A square is never negative, so (a − 1)² ≥ 0,
which means a² + 1 ≥ 2a always. The two sides
are equal only when a = 1 (there (a−1)² = 0).
A box holds at least 6 but at most 10 pencils. Write this as one combined
inequality for the count p, and list every possible value.
Show answer
6 ≤ p ≤ 10.
Both ends are included ("at least" and "at most"), so p may be 6, 7, 8, 9, or 10.
True or false, with a one-line reason:
(a) "at least 50" is the same as x > 50;
(b) if a − b < 0 then
a < b.
Show answer
(a) False — "at least 50" includes 50, so it is
x ≥ 50, not strict.
(b) True — a negative difference means a fell short of b, so a is
the smaller number.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This opening lesson targets CCSS 6.EE.B.5 (understand that solving an inequality means
finding the values that make it true) and 6.EE.B.8 (write an inequality of the form
x > c or x < c
to represent a real-world constraint, and recognize that such inequalities have infinitely many
solutions shown as a set on a number line). It also lays the groundwork for 7.EE.B.4
(using inequalities to model and solve problems). The number-line picture and the
difference-method are deliberately introduced now so that the solving work in
12.3 rests on meaning,
not memorized symbols.
The #1 misconception is confusing strict with inclusive — reading
"at least" or "at most" as >/< instead of
≥/≤ — together with misreading which way a symbol
points. The antidote, repeated until automatic: the small (open) end of the symbol points
at the smaller number, and "at least / at most" always include the boundary, so
they use ≥/≤. A quick gut-check for any
translation is to test the exact boundary value and ask whether it should count — if it should,
the inequality must be inclusive.
eastmath.com · Stage 12 · 12.1 A First Look at Inequality · Intuition before notation
eastmath.com · 12.1 A First Look at Inequality · 12.1.1 Unequal relationships all around us