Subtraction Property of Inequality
For every real n um b e r a, fa, a n d c,
if a > b, then a - c > b - c;
if a < b, then a - c < b - c.
Multiplication Property of Inequality
For every real n u m b e r a, b, and c, where c > 0,
if a > b, then ac > be;
if a < b, then ac < be.
For every real n um b er a, b, and c, where c < 0,
if a> b, then ac < be;
if a < b, then ac > be.
Division Property of Inequality
For every real n u m b e r a, b, a n d c, w h e r e c > 0,Ch ap t er 5 Linear Functions
if a < b, then § <
For every real n um b er a, b, a n d c, w h e re c < 0,
if a>b, then § <
if a < b, then § >.
Reflexive Property of Equality
For every real num ber a, a = a.
Symmetric Property of Equality
For every real n u m b e r a and b,
if a = b, then b = a.
Transitive Property of Equality
For eve ry real n u m b e r a, b, a n d c,
if a = fa a nd fa = c, th e n a = c.Transitive Property of Inequality
For every real num ber a, fa, a n d c,
if a < fa a n d fa < c, th e n a < c.
Ch ap t er 4 An Introduction to Functions
Arithmetic Sequence
The explicit form fo r the rule o f an a rith m e tic sequence is
A(n) = /4(1) + (n - 1 )d, where A(n) is th e n th te rm ,
A ( 1) is th e fir s t te rm , n is th e te r m n u m b e r, a nd
d is the com m on difference.
The recursive form fo r the rule o f an arithm etic sequence is
A(n) = A{n - 1) + d; A(1) = a, where A(n) is the nth term,
a is th e fir s t te rm , n is th e te r m n u m b e r, a n d d is th e
common difference.
Slope. = vertical change rise
" horizontal change run
Direct Variation
A d ire c t v a ria tio n is a re la tio n s h ip t h a t can be re p re s e n te d by
a fu n c tio n o f th e fo r m y = kx, where k + 0.
Slope-lntercept Form of a Linear Equation
T h e s lo p e -in te r c e p t fo r m o f a lin e a r e q u a tio n is
y = mx + fa, where m is th e s lo p e a n d fa is th e
y-intercept.
Point-Slope Form of a Linear Equation
T h e p o in t- s lo p e fo r m o f th e e q u a tio n o f a n o n v e rtic a l line
t h a t passes t h r o u g h th e p o in t (x-|, y-1) w it h s lo p e m is
y - y-i = m(x - xj).
Standard Form of a Linear Equation
T h e s ta n d a rd f o r m o f a lin e a r e q u a tio n is Ax + By = C,
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 tw o vertical lines are parallel.
Slopes o f Perpendicular Lines
Two lines are perpendicular if the product of their slopes is
-1. A vertical line and horizontal line are perpendicular.
Residual
A re s id u a l is th e d iffe re n c e b e tw e e n th e y -v a lu e o f a d a ta
p o in t a n d th e c o rre s p o n d in g y -v a lu e o f a m o d e l f o r th e d a ta
set. Y ou can f in d a re sidu al b y c a lc u la tin g y — y, where y
represents the y-value o f the data set and y represents the
corresponding y-value predicted from the model.
Ch ap t er 6 Systems o f 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 d iffe re n t slopes, the lines intersect, so
th e re is o n e s o lu tio n.
If th e lines have th e sam e slopes and d iffe re n t
y-intercepts, the lines are parallel, so there are no
solutions.
If th e lines have th e sam e slopes and th e sam e
y-intercepts, the lines are the same, so there are infinitely
many solutions.r>
OC
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