Algebra I

Sometimes the secret to an algebra problem or word problem is setting two equations equal to each other. 

Sometimes the problem will give you the equations. The classic example of this is when a problem gives you an equation for revenue and an equation for expenses, and asks for a break even point.

Example:

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?

Sometimes the secret to solving starndardized test algebra problems is just plugging in numbers.  Often the problem will give some parameters for the numbers to plug in (e.g., they must be positive or negative).  Other times, the most efficient way to solve a problem is to plug in the answer choices.  As long as you know the rules of math and algebra, you can plug in numbers and see what your answers come out to!

There are two main types of plugging in questions.  

Sometimes algebra problems will put the variable that you have to solve for in an exponent.  These problems can look intimidating but work exactly the same as other “plug in” problems.  Just find the number that makes the exponent equation true!

Getting the vocabulary right makes it easier to talk about exponent problems so:

Sometimes SAT writes equations using symbols that aren’t actually math symbols.  They use stars or hearts or other strange symbols.  The symbols have no intrinsic meaning; you just do what the problem tells you to do by following the example. 

Sometimes special symbols are just “blanks.”  You just have to replace the symbol with a number that works. 

Example:

           $\text{If }6+\bigtriangleup=18\text{, what is }\bigtriangleup\text{?}$

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.