Algebra I

Inequalities tell you something about a variable: what that variable is greater than or less than.

Compound inequalities tell you two things about a variable: two numbers that the variable is greater than or less than.

Sometimes a compound in equality will tell you what two numbers a variable lies between.  Other times, a compound in equality will give you two distinct areas of the number line that a variable could inhabit.

Inequalities are like equations.  You solve them the same way you solve equations.  However, at the end, you do not find the exact value of a variable.  Instead you learn the parameters of a value, what it's greater than or less than.

The type of equality you're solving is deteremined by the operating symbol that replaces the equal sign.  There are four inequality symbols.

You know that when you have a fraction with a square root in the denominator, you have to rationalize the denominator (essentially, multiply the fraction by a fraction, equal to one, that will cancel the radical in the original fraction).  Like so:

$\dfrac{2}{\sqrt{3}} = \dfrac{2}{\sqrt{3}}\times\dfrac{\sqrt{3}}{\sqrt{3}}=\dfrac{2\sqrt{3}}{3}$

Although sometimes, when we do a math problem, we end up with a radical in the denominator a fraction, we're not allowed to leave that radical in the denominator.

Why not?

Technically, fractions are supposed to have integers in the numerator and denominator. But mostly, it's customary to rationalize the denominator of fractions, in other words, in order for most math teacher to count an answer as "correct" there cannot be a radical in the denominator (although, interestingly, a radical in the numerator is allowed). 

An imaginary number is the square root of a negative number. How is that imaginary?  It's something that can happen when you're doing math. For instance, what if you have the equation:

$x^2=-16$

You'll square root both sides.

$\eqalign{x^2&=-16\\\sqrt{x^2}&=\sqrt{-16}\\x&=\sqrt{-16}}$

$x$ equals the square root of -16 is an answer that you could get.  But, it's not a mathematically possible number.  Why not?  What number, times itself, is equal to a negative number?

A function is a relationship between two variables.  Usually written as $f(x)$ or $g(x)$, functions are just equations (like $y = x– 4$). For example, $f(x)= x - 4$ means that for every value of $x$, $f(x)$ is $x - 4$ or $y=x-4$.

Functions tend to scare students.  For many students, as soon as they see $f(x)$, they assume that they cannot do a problem.  But functions just show how two variables are related to each other.

The distributive property says that if you multiply a term times multiple other terms (this usually looks like a term multiplied by a polynomial in parentheses) you multiply the first term by each term in the parentheses, and then combine the products. Essentially, you get rid of the parentheses by "distributing" the multiplier term to each term inside the parentheses.

When you get into long and complicated algebra equations, there are often multiple ways to solve them. 

One of the first things you should always do when working with equations, is combine like terms.  Combining like terms will not only make your equation shorter and easier to work with, but it will cut down on the number of steps you have to do when isolating a variable.

Solving an equation with fractions is the same as solving an equation without fractions.  The exact same rules apply: isolate the variables by using "opposite" operations to remove (or "undo") the terms that are attached to the variable.  "Undo" terms in reverse order of operations  (so, undo operations attached with addition and subtraction, then multiplication and division).

When you first look at it, algebra looks like math with letters.

But, what you quickly learn is that those letters, called variables, represent numbers.  You just don't know what numbers they represent yet!

In advanced algebra problems, you will solve for these variables (or the lines created by these variables).

But, at the most basic level, you can always replace variables in equations with the numbers that those variables stand for.