Basic Math and Pre-Algebra For Dummies (99 page)

Begin by multiplying –2
x
by the three terms inside the parentheses:

The expression looks worse than when you started, but you can get rid of all the parentheses by multiplying:

Now you can combine like terms:

• =
  x
  ²
  – 12
  x
  

Sometimes, expressions have two sets of parentheses next to each other without a sign between them. In that case, you need to multiply
every term
inside the first set by
every term
inside the second.

 When you have two terms inside each set of parentheses, you can use a process called FOILing. This is really just the distributive property, as I show you below. The word
FOIL
is an acronym to help you make sure you multiply the correct terms. It stands for
F
irst,
O
utside,
I
nside, and
L
ast.

Here's how the process works. In this example, you're simplifying the expression (2
x
– 2)(3
x
– 6):

1. Start out by multiplying the two First terms in the parentheses.
   

   The first term in the first set of parentheses is 2
   x,
   and 3
   x
   is the first term in the second set of parentheses: (
   2
   x
   – 2)(
   3
   x
   – 6).
   

   • F: Multiply the first terms
     :
     
2. Multiply the two
   Outside
   terms.
   

   The two outside terms, 2
   x
   and –6, are on the ends: (
   2
   x
   – 2)(3
   x
   – 6
    )
   

   • O: Multiply the outside terms: 2
     x
     (– 6) = –12
     x
     
3. Multiply the two
   Inside
   terms.
   

   The two terms in the middle are –2 and 3
   x:
   (2
   x
   – 2
   )(
   3
   x
   – 6)
   

   • I: Multiply the middle terms: –2(3
     x
     ) = –6
     x
     
4. Multiply the two
   Last
   terms.
   

   The last term in the first set of parentheses is –2, and –6 is the last term in the second set: (2
   x
   – 2
   )(3
   x
   – 6
   )
   

   • L: Multiply the last terms: –2(–6) = 12
     

Add these four results together to get the simplified expression:

In this case, you can simplify this expression still further by combining the like terms –12
x
and –6
x
:

Notice that, during this process, you multiply every term inside one set of parentheses by every term inside the other set. FOILing just helps you keep track and make sure you've multiplied everything.

 FOILing is really just an application of the distributive property, which I discuss in the section preceding this one. In other words, (2
x
– 2)(3
x
– 6) is really the same as 2
x
(3
x
– 6) + –2(3
x
– 6) when distributed. Then distributing again gives you
.

Chapter 22

In This Chapter

Using variables (such as
x
) in equations

Knowing some quick ways to solve for
x
in simple equations

Understanding the
balance scale method
for solving equations

Rearranging terms in an algebraic equation

Isolating algebraic terms on one side of an equation

Removing parentheses from an equation

Cross-multiplying to remove fractions

When it comes to algebra, solving equations is the main event.

Solving an algebraic equation means finding out what number the variable (usually
x
) stands for. Not surprisingly, this process is called
solving for x,
and when you know how to do it, your confidence — not to mention your grades in your algebra class — will soar through the roof.

This chapter is all about solving for
x.
First, I show you a few informal methods to solve for
x
when an equation isn't too difficult. Then I show you how to solve more difficult equations by thinking of them as a balance scale.

The balance scale method is really the heart of algebra (yes, algebra has a heart, after all!). When you understand this simple idea, you're ready to solve more complicated equations, using all the tools I show you in Chapter
21
, such as simplifying expressions and removing parentheses. You find out how to extend these skills to algebraic equations. Finally, I show you how cross-multiplying (see Chapter
9
) can make solving algebraic equations with fractions a piece of cake.

By the end of this chapter, you'll have a solid grasp of a bunch of ways to solve equations for the elusive and mysterious
x.

An algebraic equation is an equation that includes at least one variable — that is, a letter (such as
x
) that stands for a number.
Solving
an algebraic equation means finding out what number
x
stands for.

In this section, I show you the basics of how a variable like
x
works its way into an equation in the first place. Then I show you a few quick ways to
solve for x
when an equation isn't too difficult.

As you discover in Chapter
5
, an
equation
is a mathematical statement that contains an equals sign. For example, here's a perfectly good equation:

At its heart, a variable (such as
x
) is nothing more than a placeholder for a number. You're probably used to equations that use other placeholders: One number is purposely left as a blank or replaced by an underline or a question mark, and you're supposed to fill it in. Usually, this number comes after the equals sign. For example:

As soon as you're comfortable with addition, subtraction, or whatever, you can switch the equation around a bit:

When you stop using underlines and question marks and start using variables such as
x
to stand for the part of the equation you want to figure out, bingo! You have an algebra problem:

You don't need to call an exterminator just to kill a bug. Similarly, algebra is strong stuff, and you don't always need it to solve an algebraic equation.