Flashcards in Algebra Deck (19):
Does not have an = sign
If your equation contains a denominator,
- Put parentheses around the ENTIRE equation, and
- MULTIPLY it by the denominator
If the equation contains MULTIPLE DENOMINATORS
- Eliminate them by multiplying by the lowest common denominator (LCD)
When to cross-multiply?
If one fraction equals another
(x+1)/3 = (2x-5)/4
When working with inequalities, beware of...
Multiplying or dividing by a variable (you may need to flip the sign!)
Must consider BOTH scenarios (negative variable, positive variable)
Compound inequalities - how to isolate the variable in the middle?
Do the same thing to all 3 sides
How to solve an equation with variables AND numbers inside the absolute value brackets?
1. Isolate the absolute value brackets
2. Drop the brackets, and place +/- sign in front of the algebra (NOT in front of the answer)
3. You will then get 2 equations (2 possible answers) - solve for both of them
BUT BEWARE THAT BOTH ARE NOT ALWAYS VALID
Elimination strategy when solving 2 equations (with >1 variable)
1. Stack the equations atop one another
2. Arrange the variables in the same order
3. Add or subtract equations to ELIMINATE one of the variables
If necessary, MULTIPLY one (or both) of the equations by some value to give either variable opposite coefficients
Equations with even exponents...
...Have 2 solutions
(x-3)^2 = 25
x-3=5 and x-3=-5
Taking the root of an even exponent
Results in an absolute value
Quadratic functions - how to solve?
x^2 + 3x + 1
1. Set equal to 0
2. Make sure the x^2 term is positive
(x ) (x )
Numbers should MULTIPLY to the LAST value, and ADD to the MIDDLE value
Quadratic functions - what is a solution?
Same as the "root" - the answer to both of the factors
Factors of a quadratic equation?
Set each factor to 0 and solve.
X^2 + 7X + 10 =0 --> (X+2)(X+5) these are the "factors"
X+2=0 and X+5=0
X=-2 and X=-5 these are the "roots" aka "solutions"
(x + 2)(x + 3) = 0
How to solve for x?
DO NOT FOIL! This quadratic has already been factored. Set each factor = 0 and solve for the "roots" or "solutions."
(x + y)^2
x^2 + 2xy + y^2
X squared + 2xy + y squared
(x - y)^2
x^2 - 2xy + y^2
(x + y) (x - y)
x^2 - y^2
Plug in question
The comparison of expressions (variables can have limitless or nearly limitless values)
Goal is to prove an inconsistent relationship between the 2 quantities
Always test at least 3 values (generally -1, 0 and 1 if possible)
Remove any items shared by the two expressions
Manipulating algebra for a "plug-in" question
1. Add or subtract anything you want from both sides
2. Multiply or divide both sides by a POSITIVE value
3. Square or square root both quantities - IF they are positive