INTRODUCTION TO ALGEBRA FAST FACTS:

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||exponent||absolute value||positive and negative numbers||order of operations||
||the real number system||word problem indicators||variables, terms, and coefficients||

Raising a number to a power means to multiply that number by itself as many times as indicated by the power, or exponent. This is the exponential value of the number which is much larger than simply multiplying the two numbers.

3² = (3)(3) = 9      vs.          (3)(2) = 6

Absolute value- Absolute value can never be negative. Compute the problem as indicated inside the absolute value symbols. The result comes out of the absolute value signs as a positive. If there is a negative in front of the absolute value sign, the result would be a negative.

|-5 - 4| = |-9| = 9

-|-5 - 4| = -|-9| = -9

A negative in front of a set of parentheses changes all signs inside the parenthesis and then disappears. It can be thought of as the distribution of a negative one (-1).

-(m - 4) = -1(m - 4) = -1(m) - (-1)(4) = -m + 4

Adding like signs: When adding numbers with only positive signs, add the numbers (number of gains) and the answer is positive. When adding numbers with only negative signs, add the numbers (number of losses) and the answer is negative.

-4 + (-8) = -12

Adding unlike signs: When adding numbers with unlike signs, subtract the lower number from the higher number and use the sign of the higher number. (Please note "higher" and "lower" refer to the absolute values of each number.)

-5 + 7 = 2

5 + -7 = -2

Multiplying and Dividing like signs: Count the number of negative signs in the problem. If the number of negative signs is "even," the answer is positive. If the number of negative signs is "odd," the answer is negative.

Order of Operations:

1. Complete all operations inside parentheses

2. Simplify any expressions using exponential notation.

3. Complete all multiplication and division in order from left to right.

4. Complete all addition and subtraction in order from left to right.

Many folks use the mnemonic device:

"Please Excuse My Dear Aunt Sally" to help them remember those steps.

Please: 'P' for parentheses

Excuse: 'E' for exponents

My and Dear: 'M' and 'D' for multiplication and division

Aunt and Sally: 'A' and 'S' for addition and subtraction

Real numbers are all numbers that can be plotted on a graph. They include rational and irrational numbers.

Rational numbers are all numbers (positive and negative that are fractions or can be made into a fraction.

Irrational numbers are numbers that can not be made into exact fractions. This includes values such as "pi" and "the square root of 3".

Whole numbers are all positive real numbers starting with zero.

Integers are all real whole numbers including positive and negative numbers.

Word problem addition indicators: sum, plus, combined, included, more than.

Word problem subtraction indicators: difference, minus, take away, less than.

Word problem multiplication indicators: "of" a number, times, product.

Word problem division indicators: quotient, cutting into equal parts, sharing equal portions, equal amounts.

CAUTION!!  In word problems, "and" doesn't always mean to add. The operation word before the "and" indicates which operation sign to insert in place of the "and."

• The sum of a number and two. "Sum" means to add; therefore, replace the "and" with a plus sign (number plus two or "n + 2").

• The difference of a number and two. "Difference" means to subtract; therefore, replace the "and" with a minus sign (number minus two or "n -2").

• The product of a number and two. "Product" means to multiply; therefore, replace the "and" with a multiplication sign (number times two or "n * 2").

• The quotient of a number and two. "Quotient" means to divide; therefore, replace the "and" with a division sign (number divided by two or "n/2").

Order in addition and multiplication does not matter; however, order in subtraction and division DOES MATTER.

A variable is an unknown. We can call an unknown "x," "y," "z," etc. To find the value of the unknown, we compute the problem following order of operations. The value of the unknown depends on the other numbers in the individual problem.

A term is the number and the attached variable.  

5m - 2n

5m and -2n are TERMS

If there is no variable indicated next to a number, it is an understood variable to the zero power.

5m² - 4m + 7

The seven is understood to mean 7m⁰

Any real number or variable to the zero power equals "1."

8m⁰ = 8(1) = 8

Like terms are numbers with the same variables with the same powers.

6m² + 3n - 4m² - 3m

(The like terms are 6m² and -4m²)

Like terms can be added and subtracted. Add the numbers and attach the same variable and power as indicated in the original numbers (adding 2 apples and 3 apples equals 5 apples; therefore 2a + 3a = 5a). However, in algebra the variables stand for a number that gets multiplied by the number attached to it (no signs between letters and numbers means multiply).

Coefficients are the numbers in front of the variables. If there is no number in front of the variable, the understood coefficient is "1." If there is a negative in front of a variable with no number in front of it, the coefficient is an understood negative one.

3x² means only the x is squared. (3 times x-squared)

(3x)² means the "3" and the "x" are squared. (3x is multiplied by 3x)

A negative in front of a parentheses, brace , bracket changes all signs inside the parentheses then disappears. A positive in front of a parentheses, brace, bracket does not change any signs. A plus (positive) can simply be removed if there are no numbers in front of it.

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