How do you distribute a negative sign?
Updated
The short answer
A minus sign in front of parentheses multiplies every term inside by −1, so every sign inside flips: −(x − 4) = −x + 4. The usual slip is changing the sign of the first term and leaving the rest alone, or dropping the minus on a term inside.
What it looks like
| Wrong | Right |
|---|---|
| 5 − 2(x − 3) = 5 − 2x − 6 | 5 − 2(x − 3) = 5 − 2x + 6 |
| 3(2x − 5) = 6x + 15 | 3(2x − 5) = 6x − 15 |
In the first row, the −2 reached the x but not the −3. Putting in x = 0 shows the difference. The original gives 5 − 2(0 − 3) = 5 + 6 = 11, and so does the right line, while the wrong line gives 5 − 6 = −1. In the second row, the 3 multiplied the 5 and left its minus sign behind.
The same slip shows up when you subtract a whole expression. Subtracting 4x − 5 from x² + 3x − 5 should give x² + 3x − 5 − 4x + 5 = x² − x. If the parentheses get dropped, the subtraction only reaches the 4x, and the result is x² − x − 10. Paul Dawkins uses this example to show why the parentheses need to be written down.
In alai, an iPad app that reads your handwriting as you work, the tutor, Ally, would point at the term whose sign went wrong.
Why it happens
The minus in front of the parentheses is part of a multiplication. 5 − 2(x − 3) means 5 + (−2)(x − 3), and the −2 multiplies both terms inside: (−2)(x) = −2x and (−2)(−3) = +6.
Two things make this easy to miss. The minus sits right next to the first term, so it looks as if it belongs to that term only. And the second term needs two negatives multiplied together, which takes an extra moment of thought. Eric Schechter's page of common errors in college math names two causes of sign errors that fit here: parentheses that get lost along the way, and the belief that a minus sign always means a negative number. In fact, −x is positive whenever x is negative.
How to catch it yourself
Rewrite the subtraction as adding a negative before you multiply, so 5 − 2(x − 3) becomes 5 + (−2)(x − 3). Now there are two products to write, one for each term inside, and each one needs its own sign. Then point at the last term in your new line and ask yourself:
What does the −2 do to the −3?
You can also test the step with x = 0, as above. If the original and your new line give different numbers, a sign got lost.
Try these
- Simplify −(x − 7).
- Simplify (2a + 5) − (a − 1).
- Solve 10 − 2(x − 1) = 4.
Answers
- −x + 7
- a + 6
- x = 4, because 10 − 2x + 2 = 4 gives 12 − 2x = 4, so −2x = −8
How alai helps
In Homework help, when you pause after writing 6x + 15 = 21 under 3(2x − 5) = 21, Ally underlines the +15 and asks a question about it, for example what happens to the −5 when the 3 multiplies it. Ally then waits while you fix the line yourself. Ally also remembers the slip, so a later session can start from there.