Procedures for Intermediate Algebra

 

Order of Operations

 

 

P           Parentheses

E           Exponents

MD      Multiplication/Division (leftmost first)

AS        Addition/Subtraction (leftmost first)

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LCM and GCF

 

 

LCM	GCF
1.  Factor to primes.	1.  Same as LCM
2.  Bring down a representative of each prime.	2.  Same as LCM
3.  Bring down the most occurrences of each prime in a single denominator.	3.  Bring down least occurrences of each prime in a single number.

 

 

Exponent Problems

 

 

1.  Decide if the final answer will be + or – (count the number of negatives – not negative exponents! – and if the total is even it will be +, if odd it will be – ).

 

2.  Simplify within ( )s (sometimes there is nothing to simplify).

 

3.  Evaluate with no negative exponents.

 

How to Solve Equations

 

 

1.  Clear fractions – this is done by multiplying the entire equation by the LCM.

 

2.  Clear ( )s.

 

3.  Do + or – until the variable and its coefficient are isolated on one side of the = sign.

 

4.  Divide by the coefficient.

 

5.  If the variable has an exponent do the opposite operation (e.g. if it is squared then take the square root of both sides; if the variable is within a square root then square both sides).

 

Solving equations involving inequalities

 

1.  Solve the inequality the same as an equation.  The resulting value(s) of x is(are) considered to be critical point(s).  Place the value(s) on a number line (don't put any other numbers on this line -- just the critical point(s)).  The resulting sets of numbers are called Domains.  Test the Domains by placing an element of each Domain into the original equation and see if the answer is true.  If one element of a Domain works the entire Domain will work.  Shade in that part of the number line that is the Domain. Place the variable under the shaded area.  Put < (note the arrow is will always point left) between the variable and the Domain critical point(s). 

 

How to factor Quadratics that look like ax² + bx + c

 

1.  Factor out  the GCF.

 

2.  If it is a binomial, then it is either not factorable beyond removing the GCF or it is a difference of Squares which will look like:            a² – b² = (a – b) (a + b)

 

3.  If it is a trinomial which would look like: ax² + bx +c

 

            a.  Perfect square (a + b) ²= a² + 2ab + b² or (a – b) ²= a² – 2ab + b²             

                               

            b.  If a is a small prime, then hit or miss -- this is:  (ax +/– ___)(x +/– ___) until it works.  Remember to use the FOIL concept.

 

            c.  If a is not a prime number then use a technique I call the T-bar.  Set up a large T with f_of_i (from FOIL) = a on the left side; b  which is O + I above the vertical line in the T and l_ol_i on the right side.  Place all possible factors of a and c under them (below the bar) and match pairs until b is found.  Call one set outer (f_ol_o) and the other inner (f_il_i); remember that the larger pair will take the sign of b.

 

 

4.  Quadnomial, factor by grouping.  Take the GCF out of the first two terms and then the GCF out of the remaining two terms.  The resulting ( ) should be the same (note: sometimes you need to take out the – GCF to get the ( )s to match).