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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 2r 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.