Use the distributive property and other properties to simplify expressions and check your work.

### Distributive Property:

For any real numbers *a*,* b*, and *c*: *a*(*b* + *c*) = *ab* + *ac*.

For any real numbers *a*,* b*, and *c*: *a*(*b* – *c*) = *ab* – *ac*

The fact that two expressions are equal means that they are equal for any values of the variable. So you can check by taking a few values for the variable and making sure that those do make the two sides equal. Usually we don’t use the values 0 or 1 or 2 as the value for the variable and it is best to avoid numbers that already appear in the problem.

### Example 1

Simplify [latex]10(4x+5)[/latex].

### Example 2

Simplify [latex]-6(x+9)[/latex].

### Example 3

Simplify [latex]2(3x-7)[/latex].

### Example 4

Simplify [latex]17-5(4x-12)[/latex].

Caution: Checking your work on problems like these is not completely satisfactory, because students who make a mistake in the original solution often make a corresponding mistake in the checking process. So the fact that it checks does not provide complete confidence that the solution is correct. But it’s still a good idea to check because you’ll catch most of your mistakes.