Visual walkthrough — Inequalities — linear, solving, number line representation

This page has one job: to show you with pictures the single most important — and most forgotten — fact about linear inequalities. We will build it from the ground up, using nothing but a number line and careful looking.

Let us earn every word of that sentence.

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Step 1 — What does the symbol $<$ actually mean?

WHAT. Before we move anything around, we pin down what $2<5$ says. We read "$2<5$" as "two is less than five." But less meaning what, exactly?

WHY. If we do not know precisely what the symbol claims, we cannot check whether an operation keeps it true. Our whole flip rule will come from one clean definition, so we make it here.

PICTURE. Look at the figure. The number $2$ (chalk blue) is genuinely to the left of $5$ (chalk pink). The gap between them is drawn in pale yellow. That leftward position is all the symbol $<$ is recording.

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Step 2 — Adding the same number slides the whole line

WHAT. Take $2<5$ and add $3$ to both sides. We get $5$ and $8$. Is $5<8$ still true?

WHY. We are checking the easy rule first (the one that does not flip), so that when the flip finally appears you feel the contrast in your bones. Addition is a slide.

$\underset{\text{left number}}{\underbrace{2}}+\underset{\text{same shift}}{\underbrace{3}}<\underset{\text{right number}}{\underbrace{5}}+\underset{\text{same shift}}{\underbrace{3}}$

PICTURE. Both dots ride the same yellow arrow $+3$ steps to the right. They move together, so their left–right order is untouched. The blue dot is still left of the pink dot. The symbol stays $<$.

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Step 3 — Multiplying by a positive stretches the line but keeps order

WHAT. Now multiply $2<5$ by the positive number $2$. We get $4$ and $10$. Is $4<10$?

WHY. Multiplication is not a slide — it is a stretch away from zero. We need to see whether stretching can flip who is on the left. (Spoiler: a positive stretch cannot.)

$\underset{\text{smaller}}{\underbrace{2}}\times \underset{\text{positive scale}}{\underbrace{2}}<\underset{\text{larger}}{\underbrace{5}}\times \underset{\text{positive scale}}{\underbrace{2}}$

PICTURE. Zero (the pale-yellow anchor) stays fixed. Both dots are pushed outward to the right, but the smaller one ($2\to 4$) does not travel as far as the larger one ($5\to 10$). The blue dot stays behind. Order preserved — symbol stays $<$.

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Step 4 — Multiplying by $-1$ is a MIRROR through zero

WHAT. Here is the star of the page. Multiply $2$ by $-1$ to get $-2$; multiply $5$ by $-1$ to get $-5$. Where do they land?

WHY. Multiplying by $-1$ does not slide and does not merely stretch — it reflects every point to the opposite side of zero. Reflection is the operation that can swap left and right, so this is exactly where danger lives.

PICTURE. Watch the two curved chalk-pink arrows. The point $2$ swings across zero to land at $-2$. The point $5$ swings across zero to land at $-5$. Because $5$ started further from zero, it lands further on the negative side — that is, further left. The point that was on the left ($2$) is now on the right.

$2\stackrel{\times (-1)}{\to }-25\stackrel{\times (-1)}{\to }-5$

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Step 5 — Reading the mirrored line: the symbol MUST turn around

WHAT. We started with $2<5$ (blue left of pink). After the mirror we have $-2$ and $-5$. Which is smaller now?

WHY. "Smaller" still means "to the left" (Step 1 never changed). We just look at the mirrored line and read off the new order.

PICTURE. On the reflected line, $-5$ sits on the far left and $-2$ sits to its right. So $-5$ is the smaller one and $-2$ is the larger one: $\underset{\text{now on the RIGHT}}{\underbrace{-2}}>\underset{\text{now on the LEFT}}{\underbrace{-5}}$ The narrow point of the symbol swung from pointing at $2$ to pointing at $-5$. The symbol flipped from $<$ to $>$. Not by a rule we memorised — because the picture forced it.

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Step 6 — Edge case: what if we multiply by ZERO?

WHAT. We skipped one number: $0$. Multiply $2<5$ by $0$. We get $0$ and $0$.

WHY. A good derivation covers every sign — positive (Step 3), negative (Steps 4–5), and now the boundary between them. Zero is neither a slide nor a mirror; it is a collapse.

PICTURE. Both dots rush to the single pale-yellow point at $0$. They land on top of each other. Now $0<0$ is false, and $0>0$ is false — only $0=0$ is true. There is no left and no right left to compare.

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Step 7 — Putting it to work: solve $-4x+5>17$ visually

WHAT. We replay Worked Example 2 from the parent, but watch the line move at the flip step.

WHY. To prove the picture and the algebra agree, digit for digit.

$-4x+5>17\stackrel{-5\text{(slide)}}{\to }\underset{\text{no flip: subtraction}}{\underbrace{-4x>12}}\stackrel{÷(-4)\text{(mirror)}}{\to }\underset{\text{FLIP}}{\underbrace{x<-3}}$

• $-5$ from both sides is a slide (Step 2) → symbol stays $>$.
• $÷(-4)$ is a negative scale = mirror (Steps 4–5) → symbol turns to $<$.

PICTURE. The top line shows the solution $x>-3$ that a forgetful student writes; the bottom line shows the mirrored, correct answer $x<-3$. The chalk-pink test point $x=-4$ sits inside the correct region, and indeed $-4(-4)+5=21>17$ ✓. The blue test point $x=-2$ sits in the wrong region, and $-4(-2)+5=13\ngtr 17$ ✗.

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The one-picture summary

Three operations, three motions of the number line, one rule for each:

Operation	Motion of the line	Effect on order	Symbol
Add / subtract $k$	Slide	preserved	unchanged
Multiply / divide by positive	Stretch from 0	preserved	unchanged
Multiply / divide by negative	Mirror through 0	reversed	flip
Multiply by zero	Collapse to 0	destroyed	forbidden

Recall Feynman retelling — say it to a 12-year-old

A "less than" sign is just a photograph of who is standing on the left of the number line. When you add the same amount to both sides, both kids walk the same number of steps — the one on the left is still on the left, so the sign is fine. When you multiply by a positive number, both kids get pushed away from the middle post (zero) but stay on their own side, so again nobody swaps — the sign is fine. But when you multiply by a negative, it is like a mirror standing at zero: everybody jumps to the opposite side. The kid who was furthest away is now furthest away on the other side — so the one who was on the left is suddenly on the right! To keep the photograph honest, the "less than" sign has to spin around into a "greater than." And if you multiply by zero, both kids get squished onto the same spot and there is no left or right at all — so we simply never do that.

Connections

• Yeh sab Hinglish mein
• Linear Equations — same slides and stretches, but equations survive the mirror without flipping (both sides equal, no left/right to swap)
• Absolute Value Equations and Inequalities — reflection through zero is what $|x|$ measures
• Quadratic Inequalities — where the multiplier's sign changes across the line, forcing case-splitting
• Number Line and Real Numbers — the geometry this whole page rides on
• Systems of Inequalities · Linear Programming — many such number lines at once

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Recall Quick self-check

Multiplying an inequality by a negative number does what geometrically? ::: Reflects (mirrors) the whole number line through zero, swapping left and right, so the symbol flips. Why does adding a number never flip the sign? ::: It slides both sides by the same amount, so their left–right order is unchanged. Why must we never multiply an inequality by zero? ::: Both sides collapse to $0$; the order is destroyed and only $0=0$ remains true. Solve $-4x+5>17$. ::: $x<-3$.