lesson
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If you have \2andafriendhas$5,youhavelessmoney:2 < 5$. But what happens if you both owe triple that amount?
Suddenly you are at -\6andyourfriendisat-$15$ β now you actually have more wealth than them because you owe less. The relationship flipped entirely.
An inequality is a mathematical statement comparing two expressions that are not necessarily equal, using <, >, β€, or β₯.
πInteractive visual comparing equations and inequalities. Top card shows 'Linear Equation' 2x + 4 = 10 solved down to x = 3 with a single point on a number line. Bottom card shows 'Linear Inequality' 2x + 4 < 10 solved with identical inverse operations (-4, /2) down to x < 3 with a shaded green region to the left of an open circle at 3. Clean flat design, clear callouts showing inverse operations are identical.
Most inequality solving uses the exact same inverse operations as regular equations: adding, subtracting, multiplying, or dividing to isolate the variable.
So why did multiplying by a negative number flip our bank accounts earlier?
The Negative Rule
Take the true statement 2<5. On a number line, 5 is to the right of 2.
When you multiply both numbers by β1, they reflect across zero to β2 and β5. Now β2 is farther to the right, meaning β2>β5.
πAnimated number line diagram showing reflection across 0. At the start, two dots sit at +2 (blue) and +5 (purple) with an arrow showing 2 < 5 because 5 is further right. When a toggle 'Multiply by -1' activates, the dots trace arc paths across zero to -2 and -5. Now -2 is further right than -5, and the inequality symbol in the middle smoothly rotates from '<' to '>'. Text below: 'Multiplying by a negative reverses relative positions on the number line.'
Whenever you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.
Can we prove this algebraically without just relying on the number line picture?