Ask a hundred students to simplify 5 - (a - 2b + 3c) and a majority will write 5 - a - 2b + 3c — correct for the first term, wrong for the rest. The minus sign flipped the a and then, as far as the pen was concerned, ran out of ink. The animation below makes the bug impossible to repeat: you will literally watch the minus sign walk into the parenthesis and flip the sign of every term it passes.

The widget

A minus sign enters the parenthesis, meets each term, and flips its sign. Terms whose signs have been flipped are recoloured — red for a flip to negative, green for a flip to positive — so you can audit the minus's work term by term. The readout at the bottom writes out the expanded expression as the flips happen.

The bug: the minus sign that runs out of ink

Here is the most common wrong answer in all of beginning algebra:

5 - (a - 2b + 3c) \;\stackrel{?}{=}\; 5 - a - 2b + 3c. \quad\text{(wrong)}

The student saw the minus, touched the first term, flipped it to -a, and then copied -2b + 3c straight out of the parenthesis as if it were a quotation. The parenthesis vanished but its contents were left unchanged. This is a real bug, not a careless slip. The reason it happens is that the minus sign looks unary — like a label attached to a — rather than binary — a factor of -1 sitting in front of the whole bracket.

Fix the mental picture, and the bug goes away forever. The minus in front of the parenthesis is the exact same object as the -2 in -2(a - 2b + 3c). It just happens to be -1. And -1, like any other coefficient, distributes.

The minus is -1 \cdot

Write the expression with the hidden 1 made visible:

-(a - 2b + 3c) = (-1)\cdot(a - 2b + 3c).

Now it is a normal distribution problem — the same one you already solve when the outside factor is 3 or -2. Multiply -1 by each term inside, keeping track of each term's original sign:

(-1)(+a) + (-1)(-2b) + (-1)(+3c) = -a + 2b - 3c.

Why: multiplication is distributive over addition. The identity k(x + y + z) = kx + ky + kz holds for every real number k, including k = -1. The minus sign in front of a bracket is not a special symbol with its own rules — it is a coefficient, and it follows the coefficient rules.

Three terms in, three terms out. One sign flip per term. No term gets skipped, because the widget's minus sign does not run out of ink — -1 is not a quantity that depletes, it is a factor, and factors multiply everything they meet.

Why every sign flips, in pictures

Line the three terms up:

(a) + (-2b) + (+3c).

Now imagine the -1 factor walking past each. When -1 meets a positive, the product is negative. When -1 meets a negative, the product is positive. So:

inside the bracket after multiplying by -1
+a -a
-2b +2b
+3c -3c

That is exactly the animation: the red minus enters, each term changes colour as its sign flips, and the final line reads -a + 2b - 3c.

Worked examples — three and four terms, mixed signs

Example 1. Simplify 7 - (x - 4y + 2z).

Rewrite the bracket: -(x - 4y + 2z) = -x + 4y - 2z. Each term inside was flipped: +x \to -x, -4y \to +4y, +2z \to -2z. Now glue the 7 back on:

7 - (x - 4y + 2z) = 7 - x + 4y - 2z.

Notice what did not happen: the 7 was not touched. Only the terms inside the parenthesis feel the minus. The minus is attached to the bracket, not to the whole expression on its left.

Example 2. Simplify -(4p - 3q + r - 5s) — four terms, all different signs.

Multiply each term by -1:

+4p \to -4p, \qquad -3q \to +3q, \qquad +r \to -r, \qquad -5s \to +5s.

So -(4p - 3q + r - 5s) = -4p + 3q - r + 5s. Count the terms before and after: four in, four out. Every sign changed. If you ever end up with three signs flipped and one left alone, you have committed the exact bug this page exists to kill.

Example 3. A subtraction of a whole bracket, the JEE-exam classic: simplify (2a - b + 3c) - (a - 4b + c).

The minus in front of the second bracket flips every term inside it:

-(a - 4b + c) = -a + 4b - c.

Now add the first bracket term by term:

(2a - b + 3c) + (-a + 4b - c) = (2a - a) + (-b + 4b) + (3c - c) = a + 3b + 2c.

If you had dropped the minus after the first term, you would have written 2a - b + 3c - a - 4b + c and collected to a - 5b + 4c — a completely different answer, wrong in two of its three coefficients. The grader would have no way of knowing whether you misunderstood the distributive law or "just forgot"; the marks are the same either way.

Example 4. A nested case: simplify 3 - 2(x - (y - z)).

Start with the innermost bracket. The minus in front flips both terms inside:

-(y - z) = -y + z.

Substitute back: 3 - 2(x + (-y + z)) = 3 - 2(x - y + z). Now the -2 is a normal coefficient and distributes across all three terms:

-2(x - y + z) = -2x + 2y - 2z.

And finally 3 + (-2x + 2y - 2z) = 3 - 2x + 2y - 2z. The key move is the same each time: treat the sign in front of the bracket as a coefficient and distribute it.

What to carry forward

A minus before a parenthesis is -1 \cdot. Whenever you see one, your hand should immediately rewrite the bracket with every inside-sign flipped, before you do anything else with the expression. The animation above is the habit you are drilling: enter, flip, flip, flip, exit. If only one sign changed, you stopped too soon. If nothing changed, you never started. Every term inside, every time — that is the distributive law, and the minus is just the quiet case of it.

For the general case of distribution with any coefficient (positive, negative, variable), see the sibling visualisation Distributive Property Animated: a(b + c) Unfolds into ab + ac. For the vocabulary of terms, coefficients, and signs that lets you name each piece of an expression, return to the parent Algebraic Expressions.