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Algebra – Quick Reference Sheet

Real Numbers

Definitions:

- Natural Numbers: {1, 2, 3, …}
- Whole Numbers: {0, 1, 2, 3, …}
- Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}
- Rational Number: any number that can be written in the form $\frac{a}{b}$ where a and b are integers and b ≠ 0.
- Irrational Number: any number that cannot be written in the form $\frac{a}{b}$ where a and b are integers.
- Real Numbers: the set of numbers formed by the rational and irrational numbers.

Properties:

	Addition	Multiplication
Identity	The additive identity is 0 a + 0 = a	The multiplicative identity is 1 a + 1 = a
Inverse	The additive inverse of a is -a a + (-a) = 0	If a ≠ 0, the multiplicative inverse of a is $\frac{1}{a}$ $a(\frac{1}{a})=1$
Associative	(a + b) + c = a + (b + c)	(ab)c = a(bc)
Commutative	a + b = b + a	a • b = b • a
Distributive	a(b + c) = ab + bc	a(b – c) = ab – ac

Relations and Functions

Definitions:

- Relations: any set of ordered pairs
- Domain: first coordinates in the set of ordered pairs
- Range: second coordinates in the set of ordered pairs
- Function: a relation which pairs each element of the domain with exactly one element of the range

Notation:

Example: $f(x)=3x+7$
x corresponds to the elements in the domain
$f(x)$ is read “f of x” and corresponds to the elements in the range

Vertical Line Test:

To determine graphically whether a relation is a function, use the vertical line test

Solving Equations

Properties of Equality:

For all real numbers a, b, and c, the following properties of equality are true:

Addition Property	$If\ a=b,\ then\ a+c=b+c$
Subtraction Property	$If\ a=b,\ then\ a-c=b-c$
Multiplication Property	$If\ a=b,\ then\ a\cdot c=b\cdot c$
Division Property	$If\ a=b\ and\ c\neq 0,\ then\ \frac{a}{b}=\frac{b}{c}$

Graphing Linear Equations

Coordinate Plane:

Definitions:

slope = $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{rise}{run}$
y-intercept: point where the line crosses the y-axis; find by setting x = 0
x-intercept: point where the line crosses the x-axis, find by setting y = 0

Forms of Writing Linear Equations:

Standard Form: $Ax+By=C$
A, B, and C are real numbers and A and B are not both 0

Slope-Intercept Form: $y=mx+b$
m is the slope and b is the y-intercept

Point-Slope Form: $y-y_{1}=m(x-x_{1})$
m is the slope and $(x_{1},\ y_{1})$ is a point on the line

Methods for Graphing Linear Equations:

Plot the y-intercept first and use the slope to find additional points
Make a table of ordered pairs by choosing values for x and solving for the corresponding y values

Inequalities

Definitions:

- Inequality: a statement containing one of the symbols: <, >, ≤, or ≥
- Compound Inequality: a statement containing more than one inequality symbol

Solving Inequalities:

The Addition, Subtraction, Multiplication, and Division Properties of Equality are also true for inequalities.
* Remember: when multiplying or dividing an inequality by a negative number, you must reverse the sign (< becomes >, ≥ becomes ≤ and vice versa)

Symbol	Mark	Direction of Line
<	open circle	toward the negative numbers
>	open circle	toward the positive numbers
≤	closed circle	toward the negative numbers
≥	closed circle	toward the positive numbers

Graphing Inequalities on a Number Line:

Examples:

Graphing Inequalities on the Coordinate Plane:

Solve the inequality
Graph the boundary line
1. For the symbols <, >, ≤, and ≥ the boundary line is dashed
2. For the symbols and the boundary line is solid
Choose a point on one side of the boundary line and put its x and y values into the inequality. If they satisfy the inequality, shade on that side of the line. If not, shade on the other side of the line.

Examples:

$y<\frac{1}{2} x-1$ $y\leq x-1,\ y<-2x+2$

Radicals

Properties:

- Product ⇒ $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$
- Quotient ⇒ $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
- Radicals and exponents are inverse functions
  - $\sqrt{a^2}=|a|,\ \ \ (\sqrt{a})^{2}=a$

Solving Systems of Equations

Types of Solutions:

The solution to a pair of linear equations is an ordered pair (x, y) which represents the point of intersection of the two lines on the graph.

Answer	Number of Solutions	Graph
ordered pair	exactly one	intersecting lines
true statement	infinitely many	same line
false statement	none	parallel lines

Methods for Solving:

Substitution: solve one equation for one variable and substitute that answer into the other equation
Elimination (Linear Combination)
1. Using Addition – add the two equations together to solve for one of the variables
2. Using Subtraction – subtract one equation from the other to solve for one of the variables
3. Using Multiplication – multiply one or both equations by a constant then use addition or subtraction

Exponents

Properties:

Product of Powers	$x^{a}\cdot x^{b}=x^{a+b}$
Power of a Product	$(xy)^{a}=x^{a}y^{a}$
Quotient of Powers	$\frac{x^a}{x^b}=x^{a-b}$
Power of a Quotient	$(\frac{x}{y})^{a}=\frac{x^a}{y^a}$
Power of a Power	$(x^{a})^{b}=x^{a\cdot b}$
Negative Exponents	$x^{-a}=\frac{1}{x^a}$

Polynomials

Adding and Subtracting Polynomials:

Add and subtract like terms (those with the same exponent on the variable)

Multiplying Polynomials Using FOIL:

FOIL = First, Outer, Inner, Last

(a + b)(c + d) = ac + ad + bc + bd

Factoring:

Difference of Squares	$a^{2}-b^{2}=(a+b)(a-b)$
Perfect Square Trinomial	$a^{2}+2ab+b^{2}=(a+b)^{2}$ $a^{2}-2ab+b^{2}=(a-b)^{2}$

Completing the Square:

Original Equation	$x^{2}+bx-c=0$
Drive by the coefficient of $x^2$	$x^{2}+bx-c=0$
Move c term to other side of equation.	$x^{2}+bx=c$
Find half the x coefficient and square.	${\frac{b}{2})^{2}$
Add this term to both sides of the equation.	$x^{2}+bx+(\frac{b}{2})^{2}=c+(\frac{b}{2})^{2}$
Factor.	$(x+\frac{b}{2})^{2}=c+\frac{b^2}{4}$
Take the positive and negative square roots of both sides to find the values of x.	$x+\frac{b}{2}=\pm\sqrt{c+\frac{b^2}{4}}$

Quadratic Formula:

Used to find the roots of any quadratic equation ⇒ $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$