Download ALGEBRA 1 MID YEAR STUDY GUIDE

ALGEBRA 1 MID YEAR STUDY GUIDE
4 STEPS OF PROBLEM SOLVING
1. EXPLORE-GENERAL UNDERSTANDING OF THE PROBLEM AS TO WHAT IS
GIVEN AND NEEDED
2. PLAN- STEPS NEEDED AND OPERATIONS NEEDED TO SOLVE
3. SOLVE- CARRY OUT THE METHOD
4. EXAMINE – DOES THE ANSWER MAKE SENSE COMPARE TO AN
ESTIMATE
Order of Operations:
1.
2.
3.
4.
First do all operations that lie inside parentheses.
Next, do any work with exponents or radicals.
Working from left to right, do all multiplication and division.
Finally, working from left to right, do all addition and subtraction.
Property Laws
#1. Commutative properties
the commutative property of addition states that numbers can be added in any order and it will not change
the sum. The commutative property of multiplication states factors can be multiplied in any order without
changing the result.
Addition
5a + 4 = 4 + 5a
Multiplication
3 x 8 x 5b = 5b x 3 x 8
#2. Associative properties
The associative property of addition numbers can be grouped to form a sum in any
and still get the same answer. The associative property of multiplication numbers can
be grouped in any way to form a product and still get the same answer.
Addition
(4x + 2x) + 7x = 4x + (2x + 7x) Multiplication 2x 2(3y) = 3y(2x2)
Additive Inverse – if add a positive and negative of the same number will cancel to zero
7 + (- 7) = 0
Multiplicative Inverse- if multiply a number by its flip will equal a positive one
1/4 x 4/1 = 4/4 =1
#3. Distributive property
the distributive property involves both addition and multiplication. A longer name for
it is, "the distributive property of multiplication over addition." The term is multiplied
by terms in parenthesis, we need to "distribute" the multiplication over all the
terms inside.
2x(5 + y) = 10x + 2xy
#5. Identity property
the identity property zero added to any number is the number itself. Zero is called
the "additive identity." The identity property for multiplication, the number 1
multiplied times any number gives the number itself. The number 1 is called the
"multiplicative identity."
Addition
5y + 0 = 5y
Multiplication
2c × 1 = 2c
Substitution- a math sentence in which the variable is replaced with a number that results in the a true
sentence or in solving an equation in which an operation is replaced by a number when the operation can
not be explained by an existing property law.
( 12- 3 ) + 20/5 = 9 +4 + 13
Reflexive – when the left side = the right side of an equation x + 2 = x + 2
Symmetric – when the left side = the right by operations and uses the words if and then
If 6 + 7 = 13 then 13 = 6 + 7
Transitive- when 2 equation operations are equal and uses the if and then
If 6 + 7 = 13 and 4 + 9 = 13 Then 6 + 7 = 4 + 9
THE FACTS ABOUT INTEGERS
1. ABSOLUTE VALUE
The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the
distance of x from zero. This is why absolute value is never negative; absolute value only
asks "how far?” not "in which direction?". This means not only that | 3 | = 3, because 3 is three
units to the right of zero, but also that | –3 | = 3, because –3 is three units to the left of zero.
2. If the absolute value has a negative sign in front of the absolute value bars it means the absolute
value is still positive but was multiplied by a negative one so the solution is a negative -| 3 | = -3
3. Adding Integers
If both numbers are the same sign add the total and keep the sign the same
7 + 7 = 14
(-2) + (-3) = -5
If one number is positive and the other is negative positives and negatives will cancel each other out and
the sign is the number with the largest absolute value
7 + (-4) = + + + + + + + plus - - - - If the positive and negatives cancel 3 positives are left and 7 has
a larger absolute value than -4
(+ + + +) + + + plus (- - - -) in () means canceled and leaves 3 positives
4. Multiplying and Dividing Integers
2 negatives equals a positive even signs = positive uneven sign = negative the
the number does not have any meaning
(-5) (-2) =10 (5) (-2) = -10
(-2) (5) = -10
(5) (2) =10
12/4 = 3
absolute value of
-12/4 = -3 12/-4 = -3 -12/-4 = 3
5. Subtracting Integers
If you have a positive and subtract a positive same as basic math operations
7–2=5
If you have a negative minus a negative 2 negatives make a positive changes the
addition
7 – (-2) = 7 + 2 = 9
operation to
If you have a -7 -3 = -10 had 7 negatives and took 3 more away = -10
EQUATION IS A MATHEMATICAL SENTENCE IN WHICH A VARIABLE IS A PART OF THE
SENTENCE THAT CAN BE REPLACED WITH ONLY ONE NUMBER THAT WILL MAKE THE
SENTENCE TRUE, TO FIND THIS NUMBER USE INVERSE OPERATIONS
EX. X + 5 =12 TO SOLVE SUBTRACT 5 FROM BOTH SIDES WHICH WILL RESULT IN X = 7
X – 10 = 8 TO SOLVE ADD 10 TO BOTH SIDES WHICH WILL RESULT IN X = 18
2X = 12 DIVIDE BOTH SIDES BY 2 X= 6
INEQUALITY – A MATHEMATICAL SENTENCE THAT USES AN ORDER SIGN OF > < OR > <
SOLVED THE SAME AS EQUATIONS BUT WILL RESULT IN MORE SOLUTIONS, THE
SOLUTION MAKE NOT BE USED FOR < > ONLY BUT MAY BE USED WITH > < TO GRPAH
THE ANSWER ON A NUMBER LINE FOR < > USE AN OPEN CIRCLE FOR > < USE A CLSED
CIRCLE TO SHOW CAN USE THAT NUMBER
IN SOLVING INEQUALTIES IF THE VARIABLE HAS TO BE MULTIPLIED OR DIVIDED BY A
NEGATIVE MUST FLIP THE ORDER SIGN
Solve the inequality and graph the solution set.
*Inv. of mult. by -3 is div. by -3,
so reverse inequality sign
Graph:
*Visual showing all numbers
greater than or = -3 on the
number line
RATIO- A COMPAIRSON OF 2 NUMBERS BY DIVISION
PROPORTION- 2 RATIOS THAT HAVE EQUAL CROSS PRODUCTS, ARE
A MULTIPLE OF EACH OTHER OR WILL REDUCE TO EQUAL
FRACTIONS
MEANS-EXTREMES PROPERTY OF PROPORTIONS- PRODUCT OF
EXTREMES WILL EQUAL PRODUCT OF MEAN SO A/B = C/D THEN AD
WILL = BC : 2/3= 6/9 ( 2 x 9 ) =18 & ( 3 x 6 ) =18
PERCENT OF CHANGE
THE DIFFERENCE BETWEEN OLD AND NEW DIVIDED BY THE OLD
TIMES 100
IF AN ITEM SOLD FOR $50.00 AND NOW SELLS FOR $80 THE
DIFFERENCE BETWEEN 80 AND 50 IS 30 SO THEN DIVIDE THE 30 BY
THE ORIGINAL SALE PRICE OF $50.00 WHICH RESULTS IN A
DECIMAL OF .60 THAT TIMES 100 GIVES A 60% INCREASE
SOLVING EQUATIONS AND FORMULAS
USE THE LAWS OF ALGEBRA AND ARRANGE AN EQUATION TO
SOLVE FOR A GIVEN VARIABLE
EX. AX - B = C SOLVE FOR X ADD B TO BOTH SIDES WHICH GIVES
AX = C + B THEN DIVIDE BOTH SIDES BY A WHICH RESULTS IN
X = (C + B) / A
Weighted Averages- word problems that focus on 2 things combined to create a
third
MIXTURE PROBLEMS- WHEN 2 OR MORE PARTS ARE COMBINED TO MAKE SOMETHING
NEW
AN EQUATION WILL FORM IN THE FORM OF A + B = C THE COMBINATIONS ARE USUALLY
BY PERCENTAGE WEIGHTS OR BY A PRICE
THE INFORMATION IS ORGANIZED IN A TABLE WHICH LEADS TO THE INPUT OF AN
EQUATION
WEIGHTTED AVERAGE – A SET OF DATA IN WHICH THE SUM OF THE DATA IS DIVIDED
BY THE TOTAL OF ALL THE SUM OF THE WEIGHTS OF THE DATA
UNIFORM MOTION – ONE OBJECT MOVES AT A CERTAIN SPEED OR RATE AND USES THE
FORMULA D = R x T D = DISTANCE R = RATE T = TIME
CASE 1
IF MIXED FRUIT SELLS FOR $5.50 PER POUND, HOW MANY POUNDS OF MIXED NUTS THAT
SELLS FOR $4.75 SHOULD BE MIXED WITH 10 POUNDS OF DRIED FRUIT TO MAKE A
MIXTURE THAT SELLS FOR $4.95
WHAT HAVE
DRIED FRUIT
MIXED FRUIT
MIXTURE
AMOUNT (POUNDS)
10
X
X + 10
PRICE(POUNDS)
$5.50
$4.75
$4.95
TOTAL PRICE
5.5 (10)
4.75 ( X )
4.95 (X + 10)
EQUATION
5.5 (10 ) + 4.75 (X) = 4.95 (X + 10 )
A + B
+ C
5.5 (10 ) + 4.75 (X) = 4.95 (X + 10 )
55 + 4.75 X = 4.95 X + 49.5
4.75 X – 4.95 X = 49.5 – 55
- .20 X = 5.50
DIVIDE BOTH SIDES BY - .20 X = 27.5
CASE 2 MIXTURE PROBLEMS WITH PERCENTS- IN ORDER TO SOLVE CHANGE THE
PERCENT BACK TO THE DECIMAL FORM IN ORDER TO DETERMINE THE PART NEEDED IN
THE MIXTURE
IF A 30 % SOLUTION OF COPPER SULFATE IS NEEDED AND YOU HAVE 40 MILLILITERS OF A
25% SOLUTION, HOW MANY MILLILITERS OF A 60% SOLUTION SHOLUD BE ADDED TO
OBTAIN A 30% SOULTION?
WHAT HAVE
25% SOLUTION
60% SOLUTION
30 % SOLUTION
AMOUNT IN MILLILITERS
40 MILLILITERS
X
X + 40
TOTAL AMOUNT
.25 (40 )
.60 ( X)
.30 (X + 40 )
.25 (40) + .60 ( X) = .30 ( X + 40)
10 + .60 X = .30 X + 12
.60 X - .30 X = 12 – 10
.30 X = 2 DIVIDE BOTH SIDES BY .30 X = 6.67 MILLILITERS
Uniform Motion- trains cars etc. if go opposite directions subtract the same divide
and the distance formula is applied d = rate x time
UNIFORM MOTION
SUE DRIVES TO HER AUNT’S HOUSE AT 45 MPH RETURN TRIP IS 2 HOURS WHAT WAS HER
AVERAGE ROUND TRIP SPEED?
45(1) + (22.5 ) ( 2) = 90/3 = 30 MPH
IF A CAR AND AN AMBULANCE ARE HEADING TOWARD EACH OTHER , IF THE CAR SPEED
IS 30 MPH OR 44 FEET PER SECOND AND THE AMBULANCE SPEED IS 50 MPH OR 74 FEET
PER SECOND, IF THE VECHILES ARE 1000 FEET APART HOW MANY SECONDS BEFORE THE
CAR DRIVER WILL HEAR THE SIREN? ( THE SIREN CAN BE HEARD 440 FEET AWAY)
AMBULANCE ---------------------------------------------- CAR
1000 – 440 = 560 FEET
WHAT YOU KNOW
RATE
TIME
D=RxT
CAR
44
T
D = 44 x T
AMBULANCE
74
T
D = 74 x T
44 T + 74 T = 560
118 T = 560 DIVIDE BOTH SIDES BY 118 T = 4.75 SECONDS
GRAPHING LINEAR EQUATIONS
LINEAR EQUATION- ANY EQUATION IN WHICH THE X & Y VALUES OF A PAIR WHEN
REPLACED IN THE EQUATION BOTH SIDES ARE EQUAL AND EVERY PAIR THAT ALSO
MATCHES THE RULE WHEN CONNECTED ON A GRAPH CREATES A STRAIGHT LINE
The solutions to the equations are called the coordinates the relation x = domain & y = range
The ordered pairs can be showed in a table list or a map, the advantage of a map is a repeated x or y is only
stated once but with multiple connecting lines.
SOLVING 2 STEP EQUATIONS GOAL MOVE THE VARIABLES TO ONE SIDE AND NUMBERS
TO THE OTHER TO SOLVE FOR X WITH INVERSE OPERATIONS
Function- every x can only produce only one y in that equation so x can not be repeated
on a graph with y values and be a function of one equation the pairs must pass the vertical
line test
Slope- the change in y / change in x rise /run on a graph of a linear equation in order to
move from one pair to the next the y must go up or down and the x over in a poitive
direction or backwards in a negative direction
4 types of slope
Positive – the line moves from quadrant 3 to one and the x and the y are both positive or
negative
Negative – the line runs from quadrant 4 to two either the x or the y is negative
Zero Slope – these are the y = 3 y = -2 the y never changes the horizontal line of the y
value
Undefined slope – these are the vertical lines of the x value the x =3 x=-4
Slope and Direct Variation reflect the changes to x that create y and explain the pattern of
a graph
Direct Variation
In math when x & y are proportional in such a way that one of them is a constant
multiple of the other.
y is directly proportional to x. Y = KX always passes the point of origin
Direct Variation functions the same as slope in an equation when graphing the solutions
of an equation M & K are both constants- constants a number that does not change
If y = kx if y = 4 when x = 2 then if x = 16 what is y
Y = 2x so y = (2)(16) y = 32
This can also be calculated using ratio proportions of 4/2= y/16 so (4)(16) = 2y divide
both sides by 2 y = 32
Direct Variation Equations
2r or d
A=r2
D= rt
SLOPE- THE CHANGE IN Y OVER THE CHANGE IN X THE PATTERN OF X & Y IN AN
ORDERED PAIR TABLE
The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other
words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given
two points (x1,y1) and (x2,y2) on a line, the slope m of the line is
If use the pairs (-4 -5) & (2 7) then 7- (-5)/ 2 – (-4) = 7 +5 /2 +4 = 12/6 = 2
So the slope is 2 a rise of 2 with a run of 1
Forms of equations
1. Standard: ax + by = c ax + by – c = 0 a can not be les than zero and the equation
does not have fractions 2x + 3y = 12 or 2x + 3y -12 = 0
2. Slope Intercept – y = mx + b the m is the slope and will always be a fraction because
you must have the rise/run and every whole number has an understood one for the
denominator, the b is the y intercept
3. point slope the only purpose is to take a point and a slope of a line and form an
equation y – y1 = m ( x – x1 )
If you have 2 points on a graph ( 2, 0 ) ( 5, 4 ) to find the equation of the line first find
the slope
4 – 0 / 5 – 2 = 4/3 for the slope
Point slope y – 4 = 4/3 ( x – 5 )
To move from point slope to standard multiply both sides by 3 to remove the fraction
3y- 12 = 4 ( x – 5 ) 3y -12 = 4x -20 move the 4x to the other side so becomes
-4x + 3y = -20 + 12 so -4x + 3y = -8 a must be greater than zero so multiply or divide
both sides by -1 so the result is 4x – 3y = 8 or 4x – 3y – 8 = 0
To move to slope intercept 3y – 12 = 4x – 20 move the 12 to the other side so becomes
3y = 4x -20 + 12 so 3y = 4x -8 then divide both sides 3 so y = 4/3x -8/3
5. Parallel Equations – the lines never touch no common solutions and the equations
have the same slope so if have y = 4/3x – 8/3 the parallel equation would be y =
4/3 x with a different y intercept such as y = 4/3 x + 5
6. Perpendicular Equations – the slopes are inverse numbers and signs when
multiplied together has to equal a negative one so for y = 4/3x - 8/3 the
perpendicular equation would be y = -3/4x with any y intercept so could be
y = -3/4 x – 8/3
Function of slope – in addition to giving the pattern of change of x & y on a graph the
change of x & y also reflects a rate of change in statistical graphs by changing the
fraction of slope into a decimal and multiplying by 100 this gives the rate of change and
is used in scatter plots with a line of fit or the best line of fit
SCATTERPLOT- A GRAPH IN WHICH ONE SET OF DATA IS ON THE X AXIS AND ONE SET IS ON THE Y
AXIS, THE POINTS ARE PLOTTED BUT ARE NOT CONNECTED THE PURPOSE IS TO SEE IF THE DATA
HAS A CORRELATION THE CAUSE AND EFFECT
If have a set of data and are asked to determine the relationship and the equation of the line of fit
Use the ti-84 go to stat edit the year is the x and the birth rate is the y set the years to zero then go to 2 nd y
= which is stat plot select 1 the on the scatter plot x list L1 and the y L 2 the Zoom 9 to see the relationship
for the equation stat over to calc then 4 line of regression the equation will be the BEST FIT LINE which is
an average of the points to transfer the line y = vars 5 eq 1 then zoom to see 2nd table to see the points on
the line if asked for the line of fir means use 2 points on the graph that appear to be on the line of fit just go
to stat and enter those 2 points then stat calc and form the line of fit equation they will be close but different
Best Fit Equation y = -.193x + 16.2 Line of fit = y = -.15 x + 16.2 if use the points for 1992 and 2000
Year
Birth Rate
(per 1000)
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
2001
16.7
14.5
16.3
15.9
15.5
15.2
14.8
14.7
14.5
14.6
Source: National Center for Health Statistics, U.S. Dept. of Health and Human Services
COMPOUND INEQUALITIES
The intersection (and) and union (or )of sets of inequalities
The symbol for Union is
and the symbol for Intersection is
.
14.5
14.7
A. Solving a Compound Inequality with “AND” solution must be true to both
**4x + 5 > -7 AND 4x + 5 ≤ 25 is written like -7 < 4x + 5 ≤ 25
1. Isolate the variable in the middle.
 Distribute in the middle if possible.
 Combine like terms in the middle if possible.
 Add or subtract the number term on each side of both symbols
(middle, left, and right).
 Multiply or divide by the coefficient on each side of both symbols
(middle, left, and right).
2. If the solution contains greater than symbols, rotate the whole solution
around to get less than symbols. (This would happen when you multiply or
divide by a negative.)
3. Graph the solution.
 One of the circles goes on each number in the solution.
 A darkened bar is graphed between the two circles.Bottom of Form
2<x+2≤4
-4 < x ≤ 2
–3 ≥ 2x + 1 ≥ 5
2 ≤ x ≤ -2
This is an impossible outcome, there is no solution.
17 < 5 - 3x < 29
-8 < x < -4
Solving a Compound Inequality with “OR” solution true to only one at a time
** It’s written like 8 + 2x < 6 OR 3x - 2 > 13
1. Solve each inequality.
2. The solution must be written with two inequalities connected with “OR”.
3. Graph each inequality.
 One of the circles goes on each number in the solution.
 The darkened bar is graphed in the direction indicated by the symbol
with the number.
 If the darkened bars are going toward each other, the answer is All
Real Numbers, so you would graph a darkened bar over the entire
number line.
 If both darkened bars are going to the right, the answer is all number
> or ≥ the smallest value.
 If both darkened bars are going to the left, the answer is all number <
or ≤ the largest value.
3x + 1 < 4 OR
2x - 5 > 7
x < 1 OR x > 6
2x + 1 ≤ 7 OR -3x - 4 ≤ 2
x ≤ 3 OR x ≥ -2
This answer collapses to ALL REAL NUMBERS as
shown in the graphs below.
x-4≥3
OR
2x > 18
x ≥ 7 OR x > 9
This answer collapses to x ≥ 7 as shown in the
graphs below.
OPEN SENTENCE ABSOLUTE VALUE
An open sentence absolute value can never be = , >, or < than 0
The three forms are | x | = a positive number > than a positive number or < than a
positive number the initial is then solved in both directions such as | x + 2 | = 5 will
become x + 2 = 5 & x + 2 = -5
In the case of an inequality | x + 2 | > 4 becomes x + 2 > 4 & x + 2 < -4
| x + 2 | < 5 becomes x + 2 < 5 & x +2 > -5

Solve | x
+2|=7
To remove the absolute-value bars, split the equation into its two possible two cases,
one case for each sign:
(x + 2) = 7
x+2=7
x=5
Then the solution is x

Solve
or
or
or
–(x + 2) = 7
–x – 2 = 7
–9 = x
= –9, 5.
| 2x + 3 | < 6.
Since this is a "less than" absolute-value inequality, the first step is to clear the absolute
value according to the pattern.
| 2x + 3 | < 6
–6 < 2x + 3 < 6 [this is the pattern for "less than"]
–6 – 3 < 2x + 3 – 3 < 6 – 3
–9 < 2x < 3
–9
/2 < x < 3/2
Solving Systems on linear Inequalities:
First change each Inequality to a slope intercept form then graph each and the area
shaded by both is the solution
Boundary Line- the linear line of the inequality which is the same line as y= so points can
be found in the 2nd table function if greater or less than equal may use the points on the
line and the line will be solid if > or < than only a dotted line and can not use the points
on the line

Solve the following system:
2x – y > –3
4x + y < 5
Change to a Y Intercept form:
y < 2x + 3
y < –4x + 5
Graphing the first inequality, I get:
Drawing the second inequality, I get:
The solution is the lower region, where the two individual
solutions overlap.