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Review the key ideas, then practise whenever you’re ready.
An equation says two sides are equal (). An inequality says one side is bigger or smaller ( or ) - so instead of a single answer, you get a whole zone of answers on the number line. Solve it with the exact same balance moves as a linear equation, plus ONE golden rule when dividing by a negative.
Open circle vs Solid dot - excludes (smallest integer is ); includes (smallest integer is ).
The Negative Flip Rule - multiplying or dividing both sides by a NEGATIVE number flips to and to .
Sandwich & Absolute Value - in , do each move to all three parts. means distance from .
Interactive Inequality Number-Line Explorer
Tap each rule to see how the circle and arrow behave on the number line.
2 is the fence post, not part of the yard! The first whole number past 2 is 3.
Use an open circle ○ for < or >. Because there is no “equals” line underneath, the number 2 itself does not count.
See Why Multiplying by −1 Flips the Sign
On the right side of zero, 5 is further right than 2 (). When you multiply by , everything mirrors across zero - now −2 is further right than −5 ()!
Every step explained - why we do it, and what happens.
Why: Read it in plain words: “negative 2 times x, plus 3, is at most 11.” Unwrap x step by step just like a balanced equation.
What happens: Spot the −2 attached to x - that warns us we will divide by a negative at the end.
Why: Undo the “+ 3” first by subtracting 3 from both sides. Subtracting never flips the inequality sign.
What happens: The +3 and −3 cancel on the left, and ≤ stays pointing the same way.
11 − 3 = 8, leaving −2x ≤ 8.
Why: Divide both sides by −2 to get x alone. Because we divide by a NEGATIVE number, flip ≤ into ≥!
What happens: The left side becomes x, and the inequality arrow reverses direction.
Why: 8 ÷ (−2) = −4. Since ≥ means “greater than or equal to”, −4 itself is included.
What happens: Every number from −4 upward works, so the smallest integer solution is −4.
Why: The check: pick an easy number that obeys x ≥ −4, such as x = 0, and test it in the original inequality.
What happens: 3 ≤ 11 is true ✓ - the flipped arrow is right.
Absolute Value & Sandwich Inequalities (Optional Toolbox)
How to handle bars and three-part inequalities when you reach the higher levels.
Absolute Value Means “Distance from 1 is at most 3”
Stand at 1 on the number line. Walk 3 steps left to −2 and 3 steps right to 4:
1. Absolute Value Equation
Splits into two cases: () or ().
2. Absolute Value Inequality
Unpacks into a sandwich: , so .
3. Counting Integers in a Range
From to inclusive: integers in total. Always add so you count both endpoints!
A courier van charges a 10-coin flat pickup fee plus 4 coins for every box you send. You have 38 coins in your pocket. What is the largest whole number of boxes you can ship?
Why: Let b be the number of boxes. 4 coins per box (4b) plus the 10-coin fee must be at most 38 coins (≤).
What happens: The story becomes a two-step inequality.
Why: Subtract the 10-coin flat fee from both sides first.
What happens: You have at most 28 coins left for the boxes themselves.
Why: Divide both sides by +4. Since +4 is positive, the ≤ sign does not flip.
What happens: You can ship at most 7 boxes.
Why: The glitch: rounding up or guessing 8 boxes costs 42 coins, which goes over your 38-coin budget.
What happens: 8 boxes is too expensive - ≤ 7 is a hard ceiling.
Why: The check: 7 boxes cost 28 + 10 = 38 coins, which fits your 38-coin budget with 0 left over.
What happens: Maximum = 7 boxes ✓.
Three classic traps to watch out for:
1.Forgetting to flip the sign when dividing by a negative
Wrong: - testing gives , which is NOT
Right:Dividing by flips into : .
2.Flipping the sign just because the right side is negative
Wrong: - flipping even though you divided by positive
Right:Only look at the number you divide BY. Dividing by keeps the sign: .
3.Confusing (strict) with (inclusive)
Wrong:Saying the smallest integer that satisfies is
Right: has an open circle at ( is not allowed), so the smallest integer is .
Ready to practise? Start with a warm-up, or return to the ladder.
Learning is optional - the ladder is always open.