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Study Guide

We're already old hands at dealing with equations. Now we'll be making a lot of new kinds of equations and faux-equations, like inequalities and absolute values. In other sections it'll get even crazier: higher-order polynomials, exponential and logarithmic functions, sequences, series, the dreaded trig functions...

We just want to make sure everyone has the basics down *pat *before we dive into it all. Other acceptable ways to have the basics down are eric, jen, or rebecca. Having them down jeff, steve, or dennis are right out.

### Real Numbers

Real numbers continue to be our bread and butter in Algebra II. They include the integers (positive, negative, and zero), as well as the rational and irrational numbers. Imagine those family get-togethers; they're all so different, it's amazing they can get along at all, much less work together as a complete number system.

They do work together, though. We can add, subtract, multiply, and divide any combination of real numbers together as we please, and it all makes sense. Except division by 0: that does not work out, and it does not make sense. In a "mathematics is broken forever" kind of way.

All of this sounds familiar, right? If not, consider the possibility that someone has snuck into your brain and has stolen some of your memories. If that seems a bit far-fetched, go take a refresher, courtesy of Shmoop, instead.

### Expressions and Equations

Self-expression can take a lot of forms. We can wear crazy clothes, create sculptures dedicated to Shmoop, and gab on and on about our innermost thoughts.

In math, an

**expression**is pretty much any stand-alone mathematical formula. They're made up of constants and variables. They can't have dragons and fairies in them, though; those are just plain made-up.We can simplify expressions by performing operations—like addition or applying exponents, not brain surgery—and gathering like terms together. Just be sure to follow the order of operations, PEMDAS. Notice again that brain surgery is not on the list. Put the saw away.

**Equations**are expressions linked together by an equal sign. They declare that their two sides, no matter how complicated or different-looking they may seem, are actually equal. It doesn't matter if we're dealing with constants on each side, one variable, two, or a hundred.It also doesn't matter what those variables look like. They can be raised to a power, like

*x*^{2}, rising as a power, like 2^{x}, or mangled up in some other crazy contraption.Although the two parts of an equation are separated by an equal sign, this isn't like the Berlin Wall. Parts of the equation can pass over from one side to the other. Whether it's through addition, subtraction, multiplication, or division, we have to do it to both sides, though.

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