Simplifying

Simplifying: Putting it All Together

To simplify an expression means to tidy it so that it has as few parentheses and as few terms as possible. Ideally, it would be nice if the entire thing vanished into oblivion. That would make our lives much easier.

The way we simplify is by eliminating parentheses and combining like terms until there are no parentheses left and no like terms left to combine. Break out that vacuum cleaner.

Sample Problem

Simplify 4{3 + (x – 1)}.

First way: work from the inside out. We don't actually need those innermost parentheses. Get rid of them. They're like wisdom teeth or a Blockbuster Video gift card: completely useless and unnecessary.

4{3 + x – 1}

Now combine like terms to get:

4{2 + x}

Finally, multiply using the distributive property:

8 + 4x

Second way: work from the outside in. First, use the distributive property to get:

12 + 4(x – 1)

Then, use the distributive property again:

12 + 4x – 4

And finally, combine like terms:

8 + 4x

Okay, we're all combined out now. Break time.

Be Careful: Simplifying stuff in parentheses before you use the distributive property will often save you work. It's also a good exercise in applying the Order of Operations we learned before. Remember Please Excuse My Dear Aunt Sally? Notice that, in the second way of solving the example above, the distributive property needed to be used twice instead of once. But, you know what they say: "Using the distributive property twice is just as nice!" We're not exactly sure whom the "they" is referring to, but somebody somewhere must say it. 

Sample Problem

Show that the area of the trapezoid pictured below is given by 

First, let's chop the trapezoid into pieces (aw, don't feel bad—it has no nerve endings), and re-label some lengths. The lower base of the trapezoid has length b₂, which we can rewrite as x + b₁ + (b₂ – b₁ – x).

The area of the trapezoid is the area of the triangle on the left side, plus the area of the rectangle, plus the area of the triangle on the right side. Using the area formulas for triangles and rectangles, we get:

Ay, chihuahua! We sure would love to simplify that bad boy. Applying the distributive property gets us:

Now combine like terms, remembering that b₁h = hb₁:

Hey, look at that. Those hx terms canceled each other right out. For the final step, factor out to get:

Couldn't be more straightforward. Well, it could, but then it wouldn't be as much fun.