## Applying the Distributive Property

The distributive property is a source of many mistakes in algebra. When a negative sign, multiplication, or division is next to a set of parentheses, everything inside the parentheses will be affected.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

$(-2)(3-c)+(-2+5c-4d)-3(-d-c+1)$

$=-(6-2c)+(-2+5c-4d)-(3d+3c+3)$

$=-6-2c-2+5c-4d-3d+3c+3$

$=-8c-7c-7d+3c+3$

$=-c-7d+3c+3$

$=2c-7d+3$

.

..

Hint 1: the correct answer to the original problem is: $10c-d-11$

Hint 2: there are mistakes made between all pairs of lines except the last pair.

## Collecting Like Terms #2

The rules for simplifying an expression are often mixed up with the rules for solving equations. Remember, when working with an expression there is no “other side” to work with at the same time.

Your only options are to pay attention to the order of operations while doing nothing (do, then undo, some operation), using substitution (substitute an equivalent expression), or using the associative, commutative, or distributive properties (if they apply) to get around what the order of operations would otherwise require us to do.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

$2a-3b-4a+5b+3a$

$=2a-4a-3a+3b+5b$

$=a+8b$

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..

Hint 1: The correct answer to the problem on the first line is $a+2b$

Hint 2: Both steps above contain at least one mistake

## Absolute Value

Absolute Value symbols (two vertical bars) tell us to evaluate what is inside them, and if the result is negative change its sign before dropping the absolute value symbols.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

$|-3-4|+|5-2|-|-1+6|$

$=|-3-4|+|5-2|+|1-6|$

$=|3+4|+|5+2|+|1+6|$

$=7+7+7$

$=21$

.

..

Hint 1: the correct answer to the problem on the first line is 5.

Hint 2: more than one mistake has been made, on more than one line.