Two faces of the same idea. Left: add or subtract the same amount on both sides and the heavier side stays heavier — the inequality keeps its direction. Right: multiply both sides by a negative and the order is mirrored — the symbol flips.
An equation is fragile: do the same thing to both sides and it stays true. You learned that balancing trick in Stage 10, and it never let you down. An inequality looks like it should behave the same way — and most of the time it does. Add the same number to both sides, multiply both sides by the same number, and the statement marches along unbroken. But there is one move that betrays you, and it is the single most common mistake in all of Stage 12.
This lesson is about the legal moves — the things you are allowed to do to both sides of an inequality without changing which numbers make it true. By the end you'll be able to add, subtract, multiply, and divide both sides with confidence, you'll know exactly when the inequality symbol must flip, and you'll be able to chain and combine inequalities. Throughout, we keep the same colors: the greater side is amber, the number line and the values on it are blue, a move that keeps the direction is green, and the dreaded sign-flip is red — watch for red, because red is the trap.
12.2.1 Add or subtract the same on both sides
Picture an old-fashioned balance scale. The left pan holds a and the right pan holds b, and the left pan is heavier, so it sits lower. That tilt is the statement a>b.
Now gently set the same weight on both pans at once — say, add c grams to each side. What happens to the tilt? Nothing. You added equal weight to both pans, so the side that was heavier is still heavier by exactly the same margin. The scale doesn't budge. Take the same weight off both pans and, again, the tilt is unchanged. This is the first property, and it is the most natural one of all:
Key idea — adding keeps the direction
If a>b, then for any number c:
a + c>b + c and a − c>b − c.
The same rule holds with <, ≤, and ≥. Adding or subtracting never flips the symbol — not even when c is negative, because subtracting a positive is just adding a negative.
Let's watch it with real numbers. Start from 7>3 — plainly true. Add 4 to both sides: 11>7 ✓, still true. Now subtract 5 from both sides of the original: 2>−2 ✓, still true (and on the number line, −2 sits to the left of 2, so −2 really is the smaller one).
The whole pair of points slides the same distance left or right; the gap between them never changes, so the order never changes. Here 7>3 shifts down by 5 to give 2>−2.
Why this matters later
This is the rule that lets you move a term across an inequality. To turn x + 4 > 9 into something simpler, subtract 4 from both sides: x > 5. You'll lean on this in 12.3 every time you "transpose" a term — and you never have to worry about the symbol here.
🎮 Try it BALANCE — ADD THE SAME TO BOTH SIDES
The left pan (a) is heavier than the right pan (b). Add or remove the same weight on both sides and watch the tilt: it never changes.
Add to both pans:0
12.2.2 Multiply or divide by a positive
Adding is gentle — it slides both numbers the same distance. Multiplying is different: it stretches. But if you stretch by a positive factor, you stretch both sides away from zero in the same direction, and the bigger one stays bigger.
Here's a homely picture. Two students scored 5 and 2 on a quiz, so 5>2. The teacher decides to triple everyone's score. The first student now has 15, the second has 6. Did the ranking change? Of course not — 15>6 ✓. Tripling (a positive multiplier) treats both fairly, so whoever was ahead is still ahead. Dividing by a positive does the same in reverse: shrink both by the same positive factor and the order holds.
Key idea — × or ÷ by a positive keeps the direction
If a>b and c> 0, then:
a·c>b·c and ac>bc.
Same for <, ≤, ≥. The key words are positive multiplier — that's what protects the direction.
Tripling pushes both points further from zero by the same factor. 5>2 becomes 15>6 — the amber point stays to the right of the blue one.
Watch — "positive" is doing real work
It is tempting to think "multiplying always keeps the order." It does not. The promise above only holds while the multiplier is positive. The moment the multiplier turns negative, everything changes — which is exactly the next section.
🎮 Try it SCALE BOTH SIDES BY A POSITIVE
Start from 5 > 2. Pick a positive multiplier and watch both points stretch outward — but the amber one stays ahead.
Multiply both sides by:3
12.2.3 Multiply or divide by a negative — the flip
This is the heart of the whole lesson. Multiplying both sides by a negative number does not just stretch the picture — it reflects it through zero, like a mirror that swaps left and right. And on the number line, "to the left" means "smaller." So the number that was bigger ends up on the smaller side. The inequality flips.
Take the simplest case: 3>2, true. Multiply both sides by −1. The left becomes −3, the right becomes −2. Is −3>−2? No! On the line, −3 sits to the left of −2, so −3 is the smaller one. The true statement is
−3<−2.
The symbol turned around. Watch what would happen if you forgot:
The trap, in red
Starting from 3 > 2 and multiplying by −1, the wrong instinct is to keep the sign:
−3 > −2✗ FALSE
That statement claims −3 is bigger than −2, which is simply not so. The correct move flips the symbol:
−3 < −2 ✓
It isn't only ×(−1). Any negative factor flips. Try 6>2, multiplied by −2: the left becomes −12, the right becomes −4. Since −12 is far to the left of −4, the true statement is −12<−4 ✓ — flipped again. Dividing by a negative behaves identically, because dividing by −2 is multiplying by −½.
Key idea — × or ÷ by a negative FLIPS the direction
If a>b and c< 0, then the symbol reverses:
a·c<b·c and ac<bc.
This is the rule of the whole stage. From here on, every time you multiply or divide an inequality by a negative, the symbol must turn around. We'll flag it in red every single time.
The mirror at zero. 3>2 reflects through 0 to −3 and −2 — and now the amber point is on the left, the smaller side. The order reversed.
Move on both sides
Direction?
Add the same number
keeps
Subtract the same number
keeps
× or ÷ by a positive
keeps
× or ÷ by a negative
FLIPS
🎮 Try it THE ×/÷ FLIPPER
Start from 3 > 2. Slide the multiplier c from −3 to 3. When c is positive the order holds (green); when c is negative the two points cross and the symbol flips (red); at c = 0 both land on 0, which is why dividing by 0 is barred.
Multiplier c =−1
12.2.4 Chaining and adding inequalities
The last two properties are about combining whole inequalities, not just operating on one. The first is so natural you already use it without thinking. It's called transitivity.
Suppose you are taller than your friend, and your friend is taller than her little brother. Without measuring anyone, you know you are taller than the little brother. In symbols: if a>b and b>c, then a>c. The middle value b hands off the comparison like a relay baton.
Three heights, tallest to shortest: a>b and b>c. Reading past the middle person, a>c for free.
The second combining move lets you add two inequalities that point the same way. If a>b and another comparison says c>d, you may add them straight down:
a + c>b + d.
It makes sense: a bigger thing plus a bigger thing is a bigger total. For instance, 5 > 2 and 4 > 1, so 5 + 4 > 2 + 1, that is 9 > 3 ✓.
Watch — two things you may NOT do
Adding same-direction inequalities is safe, but two close cousins are not:
• You may not subtract them. From 5 > 2 and 4 > 1 you cannot conclude 5 − 4 > 2 − 1 (that says 1 > 1, false). Subtracting can break the direction.
• You may not multiply them unless every side is positive. Multiplying two same-direction inequalities is only guaranteed safe when all four numbers are positive.
When in doubt, add — never subtract — same-direction inequalities, and check signs before you multiply.
Putting it together
Transitivity and the addition rule are the bricks behind nearly every inequality proof you'll meet, including the famous one in 12.6. They let you build a long chain of true comparisons, link by link, each one justified by a property you've now seen.
🎮 Try it CHAIN IT UP
Two modes. Chain: set three heights a > b > c and read off a > c. Add: pick two same-direction inequalities and add them down the columns to get a + c > b + d.
Mode:
a =9
b =5
c =1
★ The big ideas, in one breath
An inequality survives almost everything you'd do to an equation: add or subtract the same number on both sides and the direction holds; multiply or divide both sides by a positive and the direction holds. The one move that betrays you is multiplying or dividing by a negative — that reflects the line through zero and turns the symbol around, so 3 > 2 becomes −3 < −2. On top of those, two combining rules: comparisons chain (a > b > c gives a > c), and same-direction inequalities can be added (but not subtracted, and not multiplied unless every side is positive).
What's next
These properties are the toolkit. In 12.3 you'll use them to solve a linear inequality almost exactly like an equation — transposing with the add/subtract rule, dividing with the ×/÷ rule — and the only fresh thing to remember will be to flip the symbol the instant you divide by a negative coefficient. After that come systems, quadratics, the basic inequality, and real-world problems — every one of them built on the moves you just learned. (If "which way does the symbol point" still feels new, revisit 12.1.)
✎ Exercises 12.2
Start from 8 > 5. Add 7 to both sides. What true statement do you get, and did the symbol change?
Show answer
15 > 12, and the symbol stays >. Adding the same number to both sides never changes the direction.
Start from 4 > −1. Subtract 6 from both sides. Write the result.
Show answer
−2 > −7 ✓. (4 − 6 = −2 and −1 − 6 = −7; on the line −7 is far left of −2, so −2 really is bigger.) Subtracting keeps the direction.
Start from 5 > 2. Multiply both sides by 3. What do you get?
Show answer
15 > 6 ✓. Multiplying by the positive number 3 keeps the symbol pointing the same way.
Start from 3 > 2. Multiply both sides by −1. Write the correct result, then state the wrong "keep the sign" answer and why it fails.
Show answer
Correct: −3 < −2 — the symbol flips because we multiplied by a negative. The tempting wrong answer is −3 > −2, which is false: −3 sits to the left of −2 on the number line, so −3 is the smaller number.
Start from 6 > 2. Multiply both sides by −2. What is the true result?
Show answer
−12 < −4. (6 × (−2) = −12 and 2 × (−2) = −4.) Any negative multiplier flips the symbol, so > becomes <.
For each move on an inequality, say keeps or flips: (a) add 5; (b) subtract 7; (c) multiply by 4; (d) divide by −3; (e) multiply by −1; (f) divide by 2.
Show answer
(a) keeps · (b) keeps · (c) keeps · (d) flips · (e) flips · (f) keeps. Only ×/÷ by a negative flips; everything else keeps the direction.
Fill in the blank so the statement is true: if −4x > 12, then x ___ −3. (Divide both sides by −4.)
Show answer
x < −3. Dividing both sides by the negative number −4 flips the symbol: 12 ÷ (−4) = −3, and > becomes <. (A peek ahead at the solving you'll do in 12.3.)
You know a > b and b > c. Can you be sure how a compares with c? Give the rule's name.
Show answer
Yes — a > c, by transitivity. The middle value b passes the comparison along: if a beats b and b beats c, then a beats c.
Given 5 > 2 and 4 > 1, add the two inequalities. State the result and confirm it is true.
Show answer
Adding down the columns: 5 + 4 > 2 + 1, i.e. 9 > 3 ✓. Same-direction inequalities may be added.
A classmate says: "From 5 > 2 and 4 > 1 I can subtract to get 5 − 4 > 2 − 1." Are they right? Show what happens.
Show answer
No. That would claim 1 > 1, which is false. You may add same-direction inequalities but you may not subtract them — subtraction can break the direction.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson develops the reasoning behind the legal operations on inequalities, aligned with CCSS 7.EE.B.4 (use variables and inequalities to solve real-world problems) and laying the foundation for A-REI.B.3 — understanding which moves preserve a solution set when solving linear inequalities. We deliberately ground each property in a concrete image first (a balance scale for add/subtract, a reflection through zero for the negative-multiply flip) before stating it in symbols, in keeping with the site's "intuition before notation" approach.
The #1 misconception is forgetting to flip the inequality symbol when multiplying or dividing by a negative — and, less often, flipping when merely adding or subtracting. The antidote is a single sharp rule worth memorizing: only ×/÷ by a negative flips; +, −, and ×/÷ by a positive all leave the direction alone. When a student is unsure, have them test with numbers (3 > 2, multiply by −1) and check on the number line that −3 really does sit to the left of −2. The "Flipper" widget above makes the reflection visible: as the multiplier crosses zero, the two points cross too, and the symbol turns red as it reverses. Reinforce that "no number" — never zero — is a legal multiplier when dividing.
eastmath.com · Stage 12 · 12.2 Properties of Inequalities · Intuition before notation
eastmath.com · 12.2 The Properties of Inequalities · 12.2.3 Multiply or divide by a negative — the flip