Algebra I

One of the most critical skills in algebra is learning to write an equation.  Being able to translate a problem -- whether a word problem or a real life problem -- into an equation opens up an entire realm of math.

So, how do you translate words into an equation?

First, some basics: 

In algebra we use variables to represent unknowns.  And, by using the tips and skills in the Creating Equations lesson, we can put together an equation from most word problems.  However, it's difficult (and sometimes impossible) to solve a single equation with two variables.  So, how can we take an equation with two unknowns and write an equation with only one variable?

Often word problems tell you how the two unknowns are related to each other, giving you a way to write one uknown "in terms of" another unknown.

One of the algebra skills that students struggle with the most is writing equations from word problems. Ironically, translating real life problems into math is one of the key ways that algebra can become useful in real life (All those times you ask yourself, why do I need to learn this?  This is why you need to learn algebra!).

Math is a precise practice and getting math right often relies upon a mutual understanding.  When I say "integer," it's important that everyone understand what an integer is (it's any non-decimal number, positive, negative, or 0).

So, not only is math vocabulary critical to executing math problems, a lot of math tests ask questions that rely explicitly on vocabulary.  So, make sure that you know these terms:

Equations are mathematical sentences.  We write equations to solve for variables that we don't know, but can predict based upon other variables. Some of the most useful -- and difficult -- math problems ask students to write or interpret equations.  What does this variable mean?  What happens to variable $n$ when variable $x$ goes up or down?

Just as there are absolute value equations, there are also absolute value inequalities.  You solve absolute value inequalities in the exact same way that you deal with absolute value equations, but with the twist that is similar to the twist involved with regular inequalities: you have to flip the inequality sign when you flip the signs of the numbers.

To solve an absolute value inequality:

You know that, in order to solve for a variable in an absolute value equation, you set the equation inside the absolute value to both the positive and negative versions of the answer.

What do you do when there are additional terms in the equation, outside of the absolute value expression?

Whenever an absolute value equation has additional terms, outside of the absolute value expression, you use algebra to get rid of those extra terms BEFORE you set the equation equal to both the positive and negative forms of the answer.

You have already learned that the absolute value of a variable tells you the value of the variable, but not whether the variable is positive or negative (and, without more information, you have to assume that it could be either).

Any time you solve for a variable whose absolute value is given, your answer will be two possible answers, as in the following equation (to review, go to the Absolute Value (with variables) lesson).

The absolute value of a number is always positive.  It represents the number's distance from 0, and distance is always positive.

However, when you are working with variables and absolute value, you have to remember that while the absolute value of a number is always positive, the original number may have been negative.

Think about this:

$\mid 6 \mid = 6$

So, if you are given $\mid x \mid = 6$, then $x$ might equal $6$.

But, remeber that:

$\mid ^-6 \mid = 6$

The absolute value of a number is the distance that number is from 0.

That means that 3 has an absolute value of 3. As you can see from the number line below, 3 is three units away from 0.