When "equals" isn't the whole story — the symbols and number-line picture for "bigger" and "smaller."
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.
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.
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.
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.
"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).
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 >.
"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 ≤.
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:
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.
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.
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.
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 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
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.
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.
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.
Read each aloud, then say which number is bigger: (a) 9 > 4 (b) −1 < 6 (c) 0 > −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)".
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.
Put each set of numbers in order from smallest to largest: (a) 3, −4, 0, −1; (b) −2, −7, −5.
Fill in > or <: (a) −5 □ −2; (b) −8 □ −10; (c) −1 □ 0.
Use the difference method to compare 57 and 710. Show the subtraction.
Use subtraction to compare 3 + 2 and 4.
Show that a² + 1 ≥ 2a for every number a, and say exactly when the two sides are equal.
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.
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.
Six questions to lock it in. Tap the answer you think is right.
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.