Strategic Math. General Review of Algebra I. With Answers. By: Shirly Boots

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1 Strategic Math General Review of Algebra I With Answers By: Shirly Boots

2 1/6 Add/Subtract/Multiply/Divide Addmoves to the right Subtract moves to the left Ex: = 6 Ex: -2 8 = - 10 Ex: 4(- 6) = -24 Ex: -16/-2 = 8 Ex: -2(-3)= 6 Ex: = 2 Ex: x(x) = x 2 Ex: x + 3x = 4x

3 1/7 Evaluate/PEMDAS P parenthesis E exponents M multiply D divide A add S subtract Evaluate means plug it in! Ex: (3 5) 2 + 7(2) -4+(2) = 14 Ex: a = 4, b = -3 ab - 5a + b 2 (4)(-3) 5(4) + (-3) = 23 Ex: a = -2, b = -5, c = 7 c 2 + ab 3a (7)2+(-2)(-5) - 3(-2) = 65

4 1/8 Combine Like Terms What are like terms? like terms have the same variables with the same exponents What is perimeter? How do I find perimeter? The distance around an object. Add up all of the sides Ex: 5a + 6ab + 7a 8b + 3ab 5a + 2b Ex: Find the perimeter 7a + 9ab 6b 2x + 5 3x -7 Ex: 2ab 2 + 3a 2 b + 7ab 4a 2 b + 5ab 2 - a 2 b + 7ab 2 + 7ab 2(2x + 5) + 2(3x 7) 4x x 14 10x - 4

5 1/12 Multiply Polynomials Distribute: multiply to everything in parenthesis after the number Ex: 5(x 2) 3(x + 7) + x(4 x) 5x 10 3x x x 2 - x 2 + 6x - 31 FOIL: double distribute - first, outside, inside, last Ex: (2x + 3)(x 5) 2x 2 10x + 3x 15 2x 2 7x 15

6 1/13 Solve 1 Variable Equations What does solve mean? To get the variable by itself. (x = #) How do I move something to the other side of the equation? Perform the opposite operation. add subtract, multiply divide, square square root Ex: 5x + 4 = 19 Ex: ½ x 7 = x = 15 ½ x = ½ ½ x = 3 x = 36

7 1/14 Solve 1 Variable Inequalities <5 Ex: 3 7x < x < x > >5 The symbol tells the direction to shade and the type of dot. Remember: when you multiply or divide by a negative, you must flip the inequality symbol < >. Ex:

8 1/20 Discount/Tax Find Cost Discount: subtract Must be a decimal Cost = Price (%)Price Tax: add Must be a decimal Cost = Price + (%)Price *change percent into decimal by moving 2 places left Ex: John buys a $ electronic device for 40% off. How much is the total costs? Ex: Sarah is charged 7.25% tax on her purchase of $ How much is the total costs? Cost = 289 (.40)(289) $173.4 Cost = (.0725)(23.89) $25.62

9 1/21 Discount/Tax: Find Original Price Discount: Cost = Price (%)Price Tax: Cost = Price + (%)Price Ex: Mike s purchase cost $ There was a 20% discount. What was the original price? = P (.2)P =.8P.8.8 $72.80 = P Ex: Alex s purchase cost $ There is an 8% tax. What was the original price? = P + (.08)P = 1.08P $75.67 = P

10 1/26 Slope from Equation Y = mx + b Ex: y = 7 2x m = - 2 slope Parallel lines have the Same Slope Perpendicular lines flip and change the sign Ex: A line parallel to y = ½ x + 3 through (0,-5) m = ½ y = ½ x 5 Ex: A line through (1, 3) perpendicular to y = - x + 4 m = y 1 = (x 3)

11 1/27 Slope from Graph Vertical lines M = undefined Horizontal Lines m = 0

12 1/28 Slope from 2 points = Ex: (2,3)(-1,7) = 4 4 = 1 m = -1 Ex: (5, -1)(5, 8) =9 0 = m = undefined, vertical line Ex: (-4,-3)(-17,-3) = 0 13 =0 m = 0, horizontal line

13 1/28 Slope from Words Key Words: each, every, per Ex: Carl gets paid $9.50 per hour at his job. m = 9.5, y = $, x = hour Ex: Susan can make 2 birdhouses every 5 days. =, y = birdhouses, x = days

14 2/2 Solve Proportions Ratio 1:3 = Steps: 1. Write as a fraction 2. Cross multiply 3. Solve Ex: = 6(x + 2) = 3(15) 6x + 12 = 45 6x = 33 x = Ex: = 2(17) = 3(x) 34 = 3x x = Ex: = 8(x + 2) = 5(x 1) 8x + 16 = 5x 5 3x = - 21 x = - 7

15 2/3 Write proportions from figures Ex: Find the missing side x x = 5(7) 2x = 35 X = 17.5 Ex: Find the missing side 4 h =h 6 3h = 4(6) 3h = 24 X = 8

16 2/4 Write Proportions from Words Ex: The exchange rate is $3 for each rubel. How many rubels is $20? $ 3x = 20 $ x = 6.67 rubels Ex: The exchange rate is 220 pesos for $1.50. How many pesos is $30? $. $ 1.5x = 30(220) 1.5x = 6600 x = 4400 pesos

17 2/9 Graph Linear Equations Y = mx + b Ex: 1 Steps: 1. Get y alone 2. Plot the y-intercept 3. Plot the slope Rise (up or down) Run (always to the right) 4. Draw the line Ex: +3

18 2/10 Linear Equations from Graph 3 4 b = 1 y = x+1 Vertical lines m = undefined b = 0 x= b = -3 y = x 3 Horizontal Lines m = 0 b= 2 y = 2

19 2/11 Linear Equations from Words Key Words: Each, per, every Y (range) comes before the key word X (domain) comes after the key word Ex: Walter gets paid $9.50 per hour. How much will he make in 22 hours? y = 9.50x y = 9.5(22) = $209 Ex: Karen makes 5 birdhouses in 2 days. How many can she make in 6 days? y = x y = (6) = 15 birdhouses Ex: The taxi driver charges $2.50 for pickup and $1.35 for each mile. How much will it cost to go 7 miles? y = x y = (7) = $11.95

20 2/17 Graph Linear Inequalities Steps: 1. Plot y intercept 2. Plot slope 3. Solid or dash line 4. Shade up or down Ex: <3+4 > Shade up > Ex: 3 Solid dash < Shade < down

21 2/18 Linear inequality from Graph Steps: 1. Where is the shading? This determines < > 2. What kind of line? This determines < > or < > 3. What is b? 4. What is the slope? 5. Plug into formula y <> mx + b Ex: Shade up, dash > m = 1, b = 1 y > x + 1 Ex: Shade down, solid < m = -3/2, b = -3 y <

22 2/18 Find the solution from the Graph Circle all solutions: Ex: (0,0) (-3,0) (-1,0) (0,5) (-4,6) (4,-6) (2,2) (5,0) Ex: (0,0) (-2,0) (-1,4) (0,-5) (-6,-6) (4,6) (2,2) (-5,0)

23 2/23 Match Graph to Equation Which one? m = -1 A. Y = x 1 B. Y= -x 1 C. Y = x + 1 D. Y = -x + 1 b=1 What do you look for? 1 st find b & eliminate answers find the sign of the slope & eliminate

24 2/24 Match words to equation Ex1: Mr. Perkins took out a loan for $800. Each month he pays $50 on the loan. subtract Ex2: Sam paid a one-time $800 connection fee for his commercial internet system. He pays $50 every month for add service. Pick the Equation for each: Y = 50x Y = x Y = 50x 800 Y = -50x 800 What do you look for? Underline numbers, find key words Y= 800x + 50

25 2/25 Match Table to Equation Ex1: -2 A. Y = -2x + 7 B. Y = 5x + 7 Ex2: +2 C. Y = 2x + 7 D. Y = -5x + 7 Ex3: +5 E. Y = -2x 7 F. Y = 5x - 7

26 2/25 Match Table to Graph Ex1: -2 A C Ex2: +2 Not here B D Ex3: +5

27 3/2 Types of Scatterplots What is the correlation? Negative, positive, none the closeness of the points and the general slope of a line drawn through the points. What do you look for? Positive negative no correlation

28 3/3 Scatterplots to Predict Domain: days Range: weight What is the weight after 83 days? 5.3 lbs. How do you find each on the graph? Graph the value you are give, draw a line that intersects with the data, find the other value. When will the animal weigh 5.8 lbs? 89 days

29 3/16 Types of Solutions for Systems Type: Type: Type: Intersection lines parallel lines same lines (or overlapping) # of Solutions: # of Solutions : # of Solutions : 1 point no solution all real numbers

30 3/17 Solve by Graphing Steps: 1. Graph first line 2. Graph second line 3. Identify the point of intersection Ex: 3 1 (-2,1)

31 3/18 Check the Answer Steps: 1. Plug x & y into 1 st equation 2. Plug x & y into 2 nd equation 3. Does each one give a true statement If either is false then the point is not a solution. Both must be true to be a solution. Ex: (4,1) Ex: (-1,2) +2=6 =3 (4) + 2(1) = 6? (4) (1) = 3? (2)? = (-1) 1 3(-1)? = 2 + 1

32 3/18 Solve Systems of Inequalities Steps: 1. Graph first line 2. Graph second line 3. Shade 1 st line 4. Shade 2 nd line 5. Identify overlapping shaded area Ex: 23 > 2

33 3/23 Solve by Substitution Steps: 1. Solve one equation for one of the variables (usually the variable with no coefficient) 2. Plug the algebraic expression into the other equation 3. Solve 4. Plug the value back into 1 st equation 5. Solve Ex: = 1 2=8 2x + (x 1) = 8 3x 1 = 8 3x = 9 x = 3 y = x - 1 y = (3) 1 y = 2 (3, 2)

34 3/24 Solve by Elimination Steps: 1. Identify the variable to eliminate (usually the one with opposite signs) 2. Multiply to make coefficients = and opposite 3. Add equations together 4. Solve 5. Plug value into original equation & solve 2=3 Ex: x + y = 3-2x + 6y = -24 7y = - 21 y = -3 2x + y = 3 2x + ( 3) = 3 2x = 6 x = 3 (3, -3) * 2

35 3/25 Write Equations for Systems Hint: 1 equation for dollars 1 for number of coins Ex: John has $4.85 in quarters and dimes. There is a total of 32 coins. How many of each coin? 10d + 25q = 485 q + d = 32 *10-10d 10q = -320 (11) + d = 32 15q = 165 d = q = 11 d = 21 Hint: 1 equation for each sentence Ex: Mark bought 2 bags of cookies and 3 dozen muffins for $9.81. Jenn bought 5 of each for $ How much for each? 2c + 3m = c + 5m = (3.93-m)+3m=9.81 c + m = m+3m=9.81 c = 3.93 m m = 9.81 c = 3.93 (1.95) m = $1.95 c = 1.98

36 3/27 Check Solutions for Word One package of chocolates costs $8. Strawberries cost $5 per pound. She cannot spend more than $80 and must have at least 6 pounds of strawberries. Which solution gets her the most for her money? A. 10 chocolate; 8 strawberries B. 8 chocolate; 3 strawberries C. 5 chocolate; 7 strawberries D. 4 chocolate; 6 strawberries What is the money equation? 8c + 5s = 80 What is another fact that we need to make a decision? > 6 strawberries How do you find the answer? Plug in each answer to money equation 8(10)+5(8) = 120 8(5)+5(7) = 75 8(4)+5(6) = 62

37 3/30 Parent Functions Linear y = x Quadratic y = x 2 Domain: R Range: R Domain: R Range: 0

38 3/31 Transformations a: (-) reflect (fraction) wide (whole #) narrow Ex: +3 k: (+) up (-) down Ex: 5 Opens down Up 3 Opens up Down 5 ½ as tall = wide

39 4/6 Factor GCF What is a GCF? The highest number that all terms have in common and the highest number of variables in common. Ex: 7x + 28 Ex: 5x 2 5x 7( x + 4) 5x(x 1) Ex: x 2 + 6x Ex: 2x 2 + 6x 8 x(x + 6) 2(x 2 + 3x 4)

40 4/7 Factor a = 1 Hints: Multiply to get 3 rd term, Add/subtract to get 2 nd term. Ex: x x + 36 Ex: x 2 9x + 20 (x+6)(x+6) (x 5)(x 4) Ex: x 2 x 12 Ex: x 2 + 4x 12 (x 4)(x + 3) (x + 6)(x 2)

41 4/8 Factor by grouping What is grouping? Working with part of the problem at a time Ex: 12a 3 9a a 15 3a 2 (4a 3) + 5(4a 3) (4a 3)(3a 2 + 5) what is the same? Ex: 6x x 2 + x + 2 6x 2 (x + 2) + 1(x + 2) (x + 2)(6x 2 + 1)

42 4/8 Factor and Solve Hint: set each piece equal to zero Ex: 4x 2 + 6x = 0 2x (2x + 3) = 0 Ex: x x + 55 = 0 (x + 11)(x + 5) = 0 2x = 0 2x + 3 = 0 x = 0 x = x + 11 = 0 x + 5 = 0 x = - 11 x = - 5

43 4/13 Domain/Range from Graph D: R R: y< -2 D: R R: R D: x < 2 R: y > -4 D: x > -2 R: y < 1

44 4/20 Domain/Range from Equation Hint: If you don t know graph it! You can use your calculator! Ex: +7 Ex: = 5 9 Domain: R Domain: R Range: y > 7 Range: y > -15

45 4/21 Domain/Range from Words Key Words: each, every, per range is before, domain is after Ex: An airplane at 5,000 feet descends at a rate of 100 feet per minute to land. Domain: minutes Range: feet Ex: Joshua earns $8 per hour. Domain: hours Range: Dollars Ex: Emma watches 20 YouTube videos every hour. Domain: hours Range: videos

46 Add/Subtract Fractions Rule: Denominator must be the same. Only the numerator adds and subtracts. Ex: () () () () Ex: () () Ex: () () () ()

47 Multiply Fractions Rule: Top times top, bottom times bottom. Always reduce the fraction first so the numbers will be smaller. Ex: Ex: Ex:

48 Divide Fractions Rule: Flip the second one over and multiply Ex: Ex: Ex:

49 Direct Variation Formula: y = kx Ex: If a is directly proportional to b, and a = 30 when b = 5, find b when a = 40. a = kb a = 6b 30 = k5 40 = 6b 6 = k 6.67 = b Ex: If r varies directly as s + 1, and r = 18 when s = 5, find r when s = 8. r = k(s+1) r = 3(s+1) 18 = k(5+1) r = 3(8+1) 18 = 6k r = 3(9) 3 = k r = 27

50 Absolute Value on Number Lines Ex: What 2 numbers are 4 away from 3? -1 and 7 Ex: What 2 numbers are 6 away from -2? -8 and 4 Ex: How far away are -3 and 5 from? distance of 8 total, half the distance is 4 4 away from 1

51 Solve Absolute Value Equations Steps: two problems in one. 1) get alone. 2) split into positive and negative parts. 3) solve each one. 4) plug the answer back in to check it. Ex: Ex: 25 1=7 3x+2=17 3x+2= x-5 =6 3x = 15 3x = -19 x-5 =3 x = 5 x = x-5 = 3 x 5 = -3 x = 8 x = 2