Why is 2(x + 3) not 2x + 3?

What it looks like

Here is the slip in the middle of solving 2(x + 3) = 14, with the correct work beside it.

Every line after the first is done correctly, which is what makes this slip hard to spot. You can see it by putting each answer back into the original equation. 2(4 + 3) = 2 × 7 = 14, so x = 4 works. 2(5.5 + 3) = 2 × 8.5 = 17, so x = 5.5 doesn't.

The same thing happens with longer expressions. In Paul Dawkins's list of common algebra errors, 4(2x² − 10) becomes 8x² − 10 when it should be 8x² − 40, and he notes that the second term gets skipped especially often when it's just a number.

In alai, an iPad app that reads your handwriting as you work, the tutor, Ally, asks about that first wrong line, because everything after it follows from it.

Why it happens

Without parentheses, 2x + 3 really does mean “2 times x, then add 3”, so your eye is used to the 2 belonging to the x alone. The parentheses change that. They say the whole group x + 3 is multiplied by 2.

It helps to read 2(x + 3) as two copies of x + 3:

(x + 3) + (x + 3) = 2x + 6

There are two x's and two 3's, so the 3 gets doubled too. The slip feels natural because 2x looks like a finished piece of algebra, while a plain number like 3 looks as if it can be copied down unchanged.

How to catch it yourself

Before you write the line after the parentheses, count the terms inside them. There are two in x + 3, so your new line needs two terms that each came from multiplying by 2. Then point at the 3 in your new line and ask yourself:

Did the 2 reach the 3 as well?

If you're not sure, test the step with a small number. With x = 1, the original is 2(1 + 3) = 8. The wrong line gives 2(1) + 3 = 5, and the right line gives 2(1) + 6 = 8. When two lines give different values for the same x, the step changed the expression.

Try these

1. Multiply out 4(x + 5).
2. Multiply out 3(2y − 7).
3. Solve 5(x + 2) = 35.

Answers

1. 4x + 20
2. 6y − 21
3. x = 5, because 5x + 10 = 35 gives 5x = 25

How alai helps

In Homework help, when you pause after writing 2x + 3 = 14 under 2(x + 3) = 14, Ally underlines the +3 and asks a question about it, for example what the 2 should multiply. The hint stops one step short, so the fix is still yours to make. If you made the same slip in an earlier session, Ally can point back to it, as in “Like last time with distributing, does the 5 in 5(x + 2) reach the 2 too?”