Subtraction Property of Inequality

For every real number a, b, and c,
if a > b , eh greater than b comma  then a − c > b − c ; eh minus c greater than b minus c semicolon
if a < b , eh less than b comma  then a − c < b − c . eh minus c less than b minus c .

Multiplication Property of Inequality

For every real number a, b, and c, where c > 0 , c greater than 0 comma
if a > b , eh greater than b comma  then a c > b c ; eh c greater than b c semicolon
if a < b , eh less than b comma  then a c < b c . eh c less than b c .

For every real number a, b, and c, where c < 0 , c less than 0 comma
if a > b , eh greater than b comma  then a c < b c ; eh c less than b c semicolon
if a < b , eh less than b comma  then a c > b c . eh c greater than b c .

Division Property of Inequality

For every real number a, b, and c, where c > 0 , c greater than 0 comma
if a > b , eh greater than b comma  then a c > b c ; eh over c , greater than , b over c , semicolon
if a < b , eh less than b comma  then a b < b c . eh over b , less than , b over c , .

For every real number a, b, and c, where c < 0 , c less than 0 comma  if
a > b , eh greater than b comma  then a c < b c ; eh over c , less than , b over c , semicolon
if a < b , eh less than b comma  then a c > b c . eh over c , greater than , b over c , .

Reflexive Property of Equality

For every real number a , a = a . eh comma eh equals eh .

Symmetric Property of Equality

For every real number a and b, if a = b , eh equals b comma  then b = a . b equals eh .

Transitive Property of Equality

For every real number a, b, and c,
if a = b eh equals b  and b = c , b equals c comma  then a = c . eh equals c .

Transitive Property of Inequality

For every real number a, b, and c, if a < b eh less than b  and b < c , b less than c comma  then a < c . eh less than c .

Chapter 4 An Introduction to Functions

Arithmetic Sequence

The form for the rule of an arithmetic sequence is A ( n ) = A ( 1 ) + ( n − 1 ) d , eh open n close equals eh open 1 close plus open n minus 1 close d comma  where A(n) is the nth term, A(1) is the first term, n is the term number, and d is the common difference.

Chapter 5 Linear Functions

Slope

slope = vertical change horizontal change = rise run slope , equals . fraction verticalchange , over horizontalchange end fraction . equals . fraction rise , over run end fraction

Direct Variation

A direct variation is a relationship that can be represented by a function of the form y = k x , y equals k x comma  where k ≠ 0 . k not equal to 0 .

Slope-Intercept Form of a Linear Equation

The slope-intercept form of a linear equation is y = m x + b , y equals m x plus b comma  where m is the slope and b is the y-intercept.

Point-Slope Form of a Linear Equation

The point-slope form of the equation of a nonvertical line that passes through the point ( x 1 , y 1 ) open , x sub 1 , comma , y sub 1 , close  with slope m is y − y 1 = m ( x − x 1 ) . y minus , y sub 1 , equals m open x minus , x sub 1 , close .

Standard Form of a Linear Equation

The standard form of a linear equation is A x + B y = C , eh x plus b y equals c comma  where A, B, and C are real numbers and A and B are not both zero.

Slopes of Parallel Lines

Nonvertical lines are parallel if they have the same slope and different y-intercepts. Any two vertical lines are parallel.

Slopes of Perpendicular Lines

Two lines are perpendicular if the product of their slopes is − 1 . negative 1 .  A vertical line and horizontal line are perpendicular.

Chapter 6 Systems of Equations and Inequalities

Solutions of Systems of Linear Equations

A system of linear equations can have one solution, no solution, or infinitely many solutions:

• If the lines have different slopes, the lines intersect, so there is one solution.
• If the lines have the same slopes and different y-intercepts, the lines are parallel, so there are no solutions.
• If the lines have the same slopes and the same y-intercepts, the lines are the same, so there are infinitely many solutions.

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