A B

-4 0 4

Math HS Prep

1) Simplify: 675÷ (6+9÷3 )675÷ (6+3 )675÷ (9 )75

2) Simplify: 3x5

x−2 5 x=3(x−2) 5 x=3 x−6 2 x=−6 x=−3

3) Simplify: 3xyx2

( 4 x6 y2

)

2y

4) Order the following from least to greatest.

a¿0.27b ¿ 14c ¿ 38d ¿ 211e¿11%

a¿0.27b¿0.25c ¿0.375d¿0.1818e ¿0.1

e ¿11%d ¿ 211b¿ 14a¿0.27c¿ 3

85) Solve each of the following equations.

a) (−5 )+9=4 b) (−9 )+5=−4c) 9+5=14d) 5+(−9 )+4=0e) 9−(−5 )=14

6) What is the approximate distance between A and B on the number line?

6−(−5 )¿11

7) Which of the following best describes the circled part of the statement? (Term, Expression or Solution)

4 x−7=5

In Algebra a term is either: * a single number, or * a variable, or * numbers and variables multiplied together.An expression - Numbers, symbols and operators (such as + and ×) grouped together that show the value of something. Example: 4x - 7A solution is a value you can put in place of a variable (such

as x) that would make the equation true.Example: 4x - 7 = 5 When we put 3 in place of x we get: 12 - 7 = 5Which is true, so x = 3 is a solution

EXPONENTS 8) Simplify: ¿

¿(27c3d18)(c4d4)Rules of exponents

¿(27c7d22)Rules of exponents

9) Simplify: (4 c4+1 )−(7c3−3 )+ (2c4+5c3 )¿4 c4+1−7 c3+3+2c4+5c3Distribute¿6c4−2c3+0c2+0c+4Standard Form

¿6 c4−2 c3+4

10)Simplify :( y¿¿2+2 y−3)−(4 y¿¿2−5 y−2)=¿¿¿

¿ y2+2 y−3−4 y2+5 y+2Distribute¿−3 y2+7 y−1Standard Form

11) Write the answer in scientific notation: (7 x103 )x (3 x104 )(7 x3 x 103+4 )21 x107

2.1 x108

12) Evaluate 7m+3mnwhenm=8∧n=107 (8 )+3 (8 ) (10 )≝. of Substitution .56+240296

13) If y=3 , then :a¿− y ¿2=b ¿(− y )¿2=¿¿−(3 )2¿(−3)2 ¿−9=9

14) If x=2∧ y=−3 , then−xy2=¿−xy2=− (2 ) (−3 )2≝. of Substitution ¿−(2)(9) ¿−18

15) Define p q by the equation p q=p2q3−3q. Then 2 3=¿2 3=2233−3 (3 )≝. of Substitution ¿108−9 ¿99

16) The number ten is raised to a power between 0 and 1. The answer has to be between which two numbers?Think: Any number raised to the power of 0 = 1So: 100=1And: 101=10Therefore the answer has to be between: 1 and 10

17) Simplify :¿16 x4 y6Rules of exponents

18) Simplify :¿27a3b6Rules of exponents

19) Simplify : (2x3 y )¿)

12 x5 y5Rules of exponents

20) Simplify :¿)¿−128 x7

21) Simplify : x2 y6 z3

x2 y2

¿ y4 z3Rules of exponents

22) Simplify :¿1

64 x3 y6Rules of exponents

23) Simplify : 3−2+2−3

¿ 132

+ 123Rulesof exponents

¿ 19+ 18Rulesof exponents

¿ 8+972

Common Denominator

¿ 1772

24) Simplify :7 x−3 y2 z−4(write using only pos. exponents)

7 y2

x3 z4

25) √50x2 (write in simplified radical form)5 x√2

26) 2 x(5+x) (write in standard form)2 x2+10 x27) (3 x¿¿3−4 x2+5)−(x2−x−7)¿ (In SF)3 x3−4 x2+5−x2+x+7

3 x3−5 x2+x+12

SOLVING EQUATIONS28) Solve this problem and indicate the rule next to

each step: 13x+28=−5

13x=−33Var one side∧¿ other side

x=−99Get rid of anyMult ¿var .29) Solve this problem and indicate the rule next to

each step: 3 x−10=2(x+3) 3 x−10=2x+6Distributex=16Var one side∧¿ other side

30) Solve this problem and indicate the rule next to each step: 4 x+5≤3+6 x

−2 x≤−2Var one side∧¿ other sidex≥1Get rid of anyMult ¿ var .

31) Solve 15+10 x>−4 x+314 x>−12Var oneside∧¿other side

x>−1214 Get rid of any Mult ¿var .

x>−67 Reduce

32) Solve this problem and indicate the rule next to each step: 3 x+17−5 x=12−(6 x+3)3 x+17−5 x=12−6 x−3Distribute−2 x+17=9−6 xCombine like terms4 x=−8Var one side∧¿other sidex=−2Get rid of any Mult ¿var .

33) Solve this problem and indicate the rule next to each step: 5 x−10=2−2 x+10 (x−3)5 x−10=2−2 x+10 x−30Distribute5 x−10=8x−28Combine like terms

18=3x Var oneside∧¿other sidex=6

34) Solve this problem and indicate the rule next to each step: x−(15 x−6 )=104x−15 x+6=104 Distribute−14 x+6=104Combinelike terms−14 x=98Var one side∧¿other sidex=−7Get rid of anyMult ¿var .

35) If 7 x+4=−19+5 x ,then 2x−14=¿7 x+4=−19+5 x Solve2 x=−23Var oneside∧¿other side

x=−232

Get rid of anyMult ¿var .

2 x−14Nowuse 2nd Equation

2(−232 )−14≝. of Substitution−23−14−37

36) Solve for x and indicate the rule next to each step: A=p+pxtA−p=pxtGet rid of any+¿−¿ var .

x= A−ppt

Get rid of any mult .¿var .

37) 4 x−6=2 ( x+3 )−44 x−6=2x+6−44 x−6=2 x+22 x=8x=4

38) 12

(4 x−8 )+2=6x+4

2 x−4+2=6 x+4

2 x−2=6 x+4−6=4 x

x=−3239) 5 ( x+3 )<7 x−3

5 x+15<7 x−318<2 x9<xx>9

40) 10<x+2<188<x<16 True and1eq .41) 12≤4 x−4 ≤816≤4 x ≤124 ≤x ≤3 False or2eq .

x≤3∨x≥442) −5 x<20x>−4

43) Solve both of the following equations in order to find the sum of the solutions for m and n. 2 (m+5 )=−254 (2−n )−2=412 (m+5 )=−25 Solve1 st equation2m+10=−25Distribute2m=−35Var one side∧¿other side

m=−352

Get rid of anyMult ¿ var

4 (2−n )−2=412nd Equation8−4 n−2=41Distribute6−4 n=41Combine like terms−4n=35Var one side∧¿other side

n=−354

Get rid of any Mult ¿var .

m+n∑ of BothEquation−352

±354

≝. of Substitution

−704

±354CommonDenominator

−1054

∨−26 14

44) You and three friends are eating a pizza with 12 pieces. Each person eats the same number of pieces. Let x represent the number of pieces each person eats. Write three equivalent algebraic equation that models this situation?

1 st : 124

=x

2nd :12=4 x

3 rd : 12x

=4

ABSOLUTE VALUES WITH INEQUALITITES

45) Graph: |2 x+3|≥1−1≥2 x+3≥1 First solve for x −3−3−3−4≥2 x≥−2Get rid of any± ¿var .−2≥x ≥−1 Falseor 2eq .x≥−1∨x ≤−2

Now plot and graph these points as filled in circles over -2 and -1

46) Graph: |2 x−3|<5−5<2 x−3<5 First solve for x +3+3+3−2<2 x<8Get rid of any± ¿var .−1<x<4True and2eq . .

47) Graph: |3 x−5|>4−4>3 x−5>4 First solve for x +5+5+51>3 x>9Get rid of any± ¿ var .

13>x>3 False or2eq .

x>3∨x< 13

QUADRATICS 48) Write in standard format: ( x−4 ) ( x+4 )

x2−16

49) Write in standard format:(4 x−3 ) ( x−4 )4 x2−19x+12

50) Write in standard format:(3 x−2 ) (4 x+1 )12 x2−5x−2

51) Write in standard format:¿(3 x+4)(3 x+4 )

9 x2+24 x+16

52) Write in standard format:¿(2 x−6)(2 x−6)

4 x2−24 x+36

53) Find all possible middle terms to make this polynomial factorable:: x2+¿¿

Think: What times what equals 10 (last term) and when added together it is a positive? (middle term)Possible Positive Numbers

(x+1)(x+10)11x (x+2)( x+5)7 x

1 & 10, 2 & 5Therefore the middle term could be: 11x or 7x

54) Find all possible middle terms to make this polynomial factorable:: x2+¿¿Think: What times what equals 20 (last term) and when added together it is a positive? (middle term)

Possible Positive Numbers(x+1)(x+20)21x (x+2)( x+10)12x (x+4 )( x+5)9 x

Therefore the middle term could be: 21x or 12x or 9x

55) Factor: 25 x2−16 y2

(5 x+4 y ) (5 x−4 y )Diff .of two squares

56) Factor: 16 x2−a2=¿(4 x+a ) (4 x−a )Diff . of two squares

57) Factor: 5 x2−3 x−2(5 x ) ( x )Signs areopposites (5 x2)(x1) (5 x+2)(x−1)Yes

58) Factor: 6 x2−13 x−5(3 x ) (2 x )Signs are opposites (3 x1)(2 x5) (3 x+1)(2 x−5)Yes

59) Factor: 8a2−17a+2¿ Maybe? (4a−2)(2a−1)NO

(8a−1)(a−2)Yes60) Factor completely: 18 x2−63 x+99 (2 x2−7 x+1 ) Factor outGCF 9(2x−5)(x−1)Signs are negative

61) Solve : ( x−5 ) ( x−3 )The "Zero Product Property" says that:If a × b = 0 then a = 0 or b = 0 (or both a=0 and b=0)The "Zero Product Property" says:If (x−5)(x−3)=0 then (x−5)=0 or (x−3)=0 Now we just solve each of those:For (x−5)=0 we get x = 5For (x−3)=0 we get x = 3And the solutions are: x = 5, or x = 3Solution: {3, 5}

62) Solve :3 x (x−4 ) (3 x+5 )=0The "Zero Product Property" says that:The "Zero Product Property" says: 3 x=0 or (x−4)=0 or (3 x+5)=0 Now we just solve each of those:For 3 x=0 we get x = 0For (x−4)=0 we get x = 4

For (3 x+5)=0 we get x = −53

Solution: {−53

, 0, 4}

63) Find the solutions: x2−x=12 x2−x−12=0 Write in Standard Form ( x−4 ) ( x+3 )=0

The "Zero Product Property" says that:Solution: {4 ,−3 }

64) Find the solutions: x2+ x=20 x2+ x−20=0 Write in Standard Form( x+5 ) ( x−4 )=0Yes

The "Zero Product Property" says that:Solution: { -5 , 4 }

65) Find the solutions to: 2 x2+11 x−21=0 (2 x−3)(x+7)

The "Zero Product Property" says that:

Solution: { -7 , 32

}

66) Find the solutions to: 2 x2−6 x−8=0

2(x¿¿2−3 x−4)=0¿ Always factor GTCF 1st if possible2 ( x−4 ) (x+1 )=0

Use "Zero Product Property"

Solution: { 4 , −1 }

67) Find the solutions: 3 x2+11 x=4

3 x2+11 x−4=0 Write in Standard Form (3 x−1 ) ( x+4 )=0

Use "Zero Product Property"

Solution: {−4 , 13

}

68) Solve: 2 x2+5 x−3=0(2 x−1 ) (x+3 )=0

Use "Zero Product Property"

Solution: {−3 , 12}

69) Solve: 2 x2+8 x+7=0 (answer in simplified radical form)

i.e. Use Quadratic Eq.

x=−b±√b2−4ac2a

x=−8±√64−4(2)(7)4

x=−8±√64−564

x=−8±√84

x=−8±√(2)(2)(2)4

x=−8±2√24

x=−2± 12 √2 or x=−4 ±√2

2

70) Find the solutions to: x2+2 x−12=−12x2+2 x=0 Write in Standard Formx (x+2 )=0Factor outGCF

Use "Zero Product Property"Solution: {−2 ,0 }

71) Simplify: x2−5 xx2−25

State limitations as well

¿ x( x−5)(x+5)(x−5) Factor 1st, x≠±5

¿x

(x+5) and x≠±5

SYSTEM OF EQUATIONS 72) Find the equation that best represents the

following word problem: In a certain freshman class, the number of girls is 83 less than twice the number of boys (b). The total number of students in that freshman class is 259. How many boys and girls are in that class?g=2b−83∧g+b=259

(2b−83 )+b=259Substitution

3b−83=259Combine like terms3b=342 b=114g=145

73) Solve the system of equations: 3 x+4 y=11x−2 y=−3

3 x+4 y=11( x−2 y=−3 )2 Use Elimination: y variable

3 x+4 y=112 x−4 y=−6 y’s are eliminated

5 x=5x=1

3 (1 )+4 y=11Substitution¿ solve x3+4 y=114 y=8Var one side∧¿ other sidey=2

Solution: {1 ,2}

74) If { 3 x+ y=10x−4 y=−1then y=¿

3 x+ y=10( x−4 y=−1 )−3 Use Elimination: x variable

3 x+ y=10−3 x+12 y=3 y’s are eliminated

13 y=13y=1

75) Solve for x and y: 2 x+4 y=12−2 x+ y=8

Use Elimination: x variable

5 y=20y=4

2 x+4(4)=12Substitution2 x+16=122 x=−4x=−2Solution: {−2 ,4 }

76) Solve for a and b: a−b=5

3a+2b=−25

Use Elimination: b variable

a−b=5(2)3a+2b=−25

2a−2b=103a+2b=−25

5a=−15a=−3

(−3 )−b=5−b=8b=−8Solution: {−3 ,−8 }

77) Find the x-coordinate of the system: 3 x+3 y=4x−3 y=1

3 x+3 y=4x−3 y=1 Use Elimination: y variable

4 x=5Get rid of anyMult ¿ var

x=54

78) If {x+ y=6x= y+2then y=¿

( x )+ y=6Use Substitution( y+2 )+ y=6Substitution2 y+2=6Combine like terms2 y=4Get rid of any Mult ¿vary=2

LINERAR EQUATIONS: (SLOPE, MPS, SI, SF) 79) Find the slope of the line that passes through

(4, 7) and (1, 3)

Slope= riserun

Slope= Δ yΔ x

Slope=( y1 )−( y2 )(x1 )−(x2 )

Slope= (7 )− (3 )(4 )−(1 )

Slope=43

80) Find the slope of the line that passes through (-4, 7) and (0, -6)

Slope= (7 )−(−6 )(−4 )−(0 )

Slope= 13−4

81) Find the slope and y-intercept of the line whose

equation is: y=−32x+4

Slope Intercept (SI ) : y=mx+b

slope=−32 yint=4

82) Find the slope and y-intercept of the line whose equation is: 2 x+6 y=12

6 y=−2x+12

y=−26x+ 12

6

y=−13x+2

slope=−13 yint=2

83) Write the equation in all three forms (MPS, SI, and SF) of the graph below?

Slope=2 , pt 1=(4 ,5 ) , pt2=(6 ,9 ) MPS : y=m (x−x1 )+ y1 MPS : y=2 ( x−4 )+5 (Use pt 1)

SI : y=mx+bSI : y=2 x−8+5Distribute MPS SI : y=2 x−3SF : Ax+By=CSI :−2x+ y=−3Use SI :Get x on¿SF :2x− y=3

84) Where does the graph of x−4 y+8=0 cross the y-axis? Crossing at the y-axis means that: x = 0

(0 )−4 y+8=0 Substitution : x=0 −4 y+8=0 8=4 yGet rid of any Mult ¿var

y=2 y∫ .=(0 ,2)

85) Find the distance between P(-8, -3) and Q(7, 5) (If necessary use radical form)a2+b2=c2PythagoreanTheoremc=√a2+b2

Distance=√a2+b2Distance=√(x1−x2)

2+( y1− y2)2

Distance=√(−8−7)2+(−3−5)2

Distance=√(−15)2+(−8)2

Distance=√225+64Distance=√289Distance=17

86) Find the distance between P(-4, 6) and Q(-2, -8) (If necessary use radical form)PythagoreanTheorem

a2+b2=c2

c=√a2+b2

Distance=√a2+b2Distance=√(x1−x2)

2+( y1− y2)2

Distance=√(−4−−2)2+(6−−8)2

Distance=√(−4+2)2+(6+8)2

Distance=√(−2)2+(14)2

Distance=√4+196Distance=√200Distance=√(2)(100)

Distance=10√2

87) Write the equation in all three forms (MPS, SI, and SF) of the graph below?

Slope=−2 , pt 1=(−3 ,8 ) , pt 2=(5 ,−8 )

MPS : y=m (x−x1 )+ y1 MPS : y=−2 ( x+3 )+8 (Use pt 1) SI : y=mx+bSI : y=−2x−6+8DistributeMPS SI : y=−2x+2SF : Ax+By=C

SF :2 x+ y=2

88) Find the equation in all three forms (MPS, SI, and SF) of the line containing the point (-3, 5) and having a slope: 4

MPS : y=4 ( x+3 )+5

SI : y=4 x+12+5Distribute MPS SI : y=4 x+17SF :−4 x+ y=17Use SI :Get x on¿SF :4 x− y=−17

89) Find the equation in all three forms (MPS, SI, and SF) of the line containing the points (-3, -12) and (-1, -20)

Slope=( y1 )−( y2 )(x1 )−(x2 )

First find the slope

Slope= (−12 )−(−20 )(−3 )−(−1 )

Slope=−12+20−3+1

Slope= 8−2

Slope=−4

MPS : y=m (x−x1 )+ y1 MPS : y=−4 (x+3 )−12

SI : y=mx+bSI : y=−4 x−12−12Distribute MPS SI : y=−4 x−24SF : Ax+By=CSF :4 x+ y=−24

90) Find the equation in all three forms (MPS, SI, and SF) of the line passing through (5, 2) and (6, 4)

Slope= (2 )−(4 )(5 )−(6 )

Slope=−2−1

Slope=2

MPS : y=2 ( x−5 )+2 y=2x−10+2 SI : y=2 x−8−2 x+ y=−8SF :2 x− y=8

91) Find the equation in all three forms (MPS, SI, and SF) of the line passing through (-4, 8) and (2, -4)

Slope= (8 )−(−4 )(−4 )−(2 )

Slope= 12−6

8

8 9

20

x y

w z

Slope=−2

MPS : y=−2 ( x+4 )+8

y=−2x−8+8 SI : y=−2 xSF :2x+ y=0

92) At a bake sale you sold pies for $10 and cakes for $15. If you sold 42 items and your total sales were $480, then how many of each item did you sell?10 P+15C=480P+C=42 ( -10)

10 P+15C=480−10 P−10C=−420 5C=60 C=12cakes soldP+12=42 P=30 pies sold

93) Describe the pair of lines represented: same line, parallel, perpendicular or just intersecting lines?

a) y=3 x+7y=3x+2 d)

y=5 x+5y=10 x+5

b) 2 y=4 x−16y=2x−8 e)

y=2 x+1y=−½ x−5

c ¿ y=xy=3 f)

y=9x=5

a) Both have a slope of 3, but have different y-int. Therefore the equations are parallel. b) The 1st equation can be reduced to 2nd by two. Therefore the equations are the same line. c) The 1st eq. has a slope of 1. The 2nd equation is a vertical line, with an undefined slope. Therefore the equations are intersecting. d) Both have different slopes of 5 and 10. The slopes are not negative reciprocals. Therefore the equations are intersecting. d) Both have different slopes of 2 and -½. The slopes are negative reciprocals. Therefore the equations are perpendicular. f) The 1st eq. is a horizontal line, with a slope = 0 The 2nd eq. is a vertical line, with an undef. slope. Therefore the equations are perpendicular.

94) Find the equation in all three forms (MPS, SI, and SF) of the perpendicular bisector between K(-5, -7) and L(1,5)Perpendicular=negative reciprocal slope

Slope=( y1 )−( y2 )(x1 )−(x2 )

First find the slope

Slope= (−7 )− (5 )(−5 )− (1 )

Slope=−12−6

Slope=2

Perpendicular Slope=−12

Bisector=midpoint between K∧Lx1+x22

∧ y1+ y2

2−5+12

∧−7+5

2−42

∧−2

2Midpoint (−2 ,−1)

MPS : y=−12

(x+2 )−1

y=−12x−1−1

SI : y=−12x−2

2 y=−x−4SF : x+2 y=−4

95) A boy is mowing a rectangular lawn 40 ft. long and 30 ft. wide. He has cut all of it except for a rectangle that is 20 ft. long and 15 ft. wide. What fractional part of the lawn remains uncut?uncut areatotal area

(20)(15)(40)(30)

(1)(1)(2)(2)

14

96) If you choose a random point on a side of the trapezoid, what is the probability that it is on WZ?

distanceWZtotal perimeter distance

2045

Probabiliby=49

12

15

15

6

12

15

12 ft

20 ft

3

46

97) The perimeter of a rectangle is 28in. If its length is 9 in., find its area. Perimeter rectangle=2(l+w)28=2 ( l+w )28=2 (9+w )Substitution28=18+2w Distribute10=2wVar oneside∧¿other sidew=5Areaof rectangle=(l)(w)Areaof rectangle=(9)(5)Areaof rectangle=45¿2

98) Find the area of a right triangle with hypotenuse 15 in. and altitude 12 in (to the hypotenuse).

Areaof triangle=(b)(h)2

Areaof triangle=(15)(12)2

Areaof triangle=90¿2

99) Find the area of the trapezoid.

Areaof trapezoid=( b1+b22 )(h)PythagoreanTheorem : a2+b2=c2

a=√c2−b2h=√122−62h=√144−36h=√108h=√(3)(36)h=6√3

Areaof trapezoid=( 15+212 )(6 √3)

Areaof trapezoid=(18)(6√3)Areaof trapezoid=108√3

100) Find the area of a circle with circumference 32π . Circumference of ¿̊2π r32π=2π r Substitution ¿ solve for r32=2 rr=16Areaof ¿̊ π r2

Areaof ¿̊ π (16)2SubstitutionAreaof ¿̊256 π

101) Find the volume of a cylinder that has a diameter of 12 in. and a height of 15 in. Volumeof cylinder=(Area of Base)(height )Volumeof cylinder=(π r2)(h)V=π r2hV=π (62) (15 ) SubstitutionD=12 r=6Volume=540π ¿3

102) Find the area and perimeter of rectangle ABCD.

BC=12 diameter=20AD=12 radius=10OD=6 OC=10CD=√102−62 CD=√100−36CD=√64CD=8Areaof rectangle=(b)(h)Areaof rectangle=(12)(8)Areaof rectangle=96 ft2

Perimeter of rectangle=2(b+h) Perimeter of rectangle=2(12+8) Perimeter of rectangle=2(20) Perimeter of rectangle=40 ft .

103) Find the volume of the prism.

Volumeof tri . prism=(Areaof Base)(height )

Volumeof tri . prism=((b)(h1)2 )(h2)V=((4)(3)2 )(6)SubstitutionVolume=36u3

104) Suppose ABCΔ PQR with AB:PQ = 5:9 ΔWhat is the ratio of the area of ABC to PQR?Δ Δ

If two triangles are similar, each side will have the same proportional relationship. Area is squared units, so the ratio of the areas is:

20x

14x

y

312

15

y

3

y

12

A

8

BC 15

Ratio of the areaof ΔABC ¿ΔPQR=( 5292 )¿ 2581

∨25 :81

105) Find the area of the shaded region (a regular hexagon inscribed in the circle with a radius of 10

Area Shaded=Area ˚−Area of Hexagon

¿ π r2−(Apothem)(Perimeter)2

¿ π (10)2−(5√3)(6)(10)2

Area Shaded=100π−150√3

106) Find the exact value of x.

Special¿Triangle :30¿o60o90o

x=10√3

107) Find the exact value of x.

Special¿Triangle :45¿o45o90o

x=7√2

108) Find the exact value of y:

Special¿Triangle :Ratio3 Δ

Can’t use here

These last two triangles havemeasures that canbeused ¿determine the value of y .

Ratio= singledouble

Δ1= singledouble

Δ1∧Δ2 : y12

= 3y

y2=(12)(3)y2=36y=√36y=6

109) Given A (0, 2), B (5, 5), and C (7, 2). Reflect ABC Δover the x-axis. Give the coordinates of the vertices of the image.Reflecting over the x-axis will make the x-value stay the same, but the y-value will be the opposite.OriginalValues :A (0 ,−2 )B (5 ,−5 )C (7 ,−2)

NewValues : A' (0 ,−2 )B' (5 ,−5 )C ' (7 ,−2 )

110) (a) Find the trig ratios of sin, cos and tan of angle A, (b) Next do the same for angle B

The trigonometric ratios are special measurements of a right triangle. Since we have to find the three primary trig ratios it is necessary to find the hypotenuse (side c), which is length AB

PythagoreanTheorem : a2+b2=c2

c=√a2+b2c=√82+152c=√64+225

21

x

23

34

56

7

1 28

B

E

C

D

c=√289c=17

sin A= OppositeHypotenuse

=1517 sin B= 8

17

cos A= AdjacentHypotenuse

= 817 cosB=15

17

tan A=OppositeAdjacent

=158 tanB= 8

15

111) Use a trig ratio to find the value of y.

cos62o=21y cos62o= Adjacent

Hypotenuse( y ) cos62=21Get var∈numerator

( y )cos 62cos62

= 21cos 62

Get rid of anyMult ¿ var

y=44.731Use your scientific calculator112) Find the value of x.

Triangle Midpoint Theorem - In any triangle, a segment joining the midpoints of any two sides will be parallel to the third side and half its length. x=232 or x=11.5

113) Find the value of x.

Alternate Exterior angles - Pairs of angles on opposite sides of the transversal but outside the two lines are called Alternate Exterior Angles. If the two lines are parallel the angles are congruent.3 x−9=114 Solve3 x=123Var oneside∧¿other side

x=1233Get rid of any Mult ¿var .

x=41114) State the pairs of same-side interior angles.

Same Side Interior angles - Pairs of angles on the same side of the transversal but inside the two lines. If the two lines are parallel the angles are supplementary.SSI=2∧3 and SSI=6∧7

115) Find the value of x and y that will make this a parallelogram.

If the two lines are parallel the Same Side Interior angles are supplementary.

( x+30 )+2 x=180Combine like terms3 x+30=180Var one side∧¿other side3 x=150Get rid of anyMult ¿ var . x=50If the two lines are parallel the Same Side Interior angles are supplementary.

(3 y−20 )+2x=180 Substitute(3 y−20 )+2(50)=1803 y−20+100=180Combinelike terms3 y+80=180Var one side∧¿other side3 y=100Get rid of any Mult ¿var .

y=1003

x , y=(50 , 1003 )116) In the diagram below, mBCEis144 ° Find

mECD .

mBCD+mDCE=mBCE(3 x+12 )+ (2x+32 )=144 Substitution5 x+44=144Combine like terms5 x=100Var oneside∧¿other sidex=20Get rid of any Mult ¿varmDCE=mECD=2x+32mECD=2 (20 )+32SubstitutionmECD=72

117) Find the value of y.

y

Triangle Sum Theorem - The sum of the interior angles of a triangle is 180 degrees

y+ y+(2 y−20 )=180 Δ∑Theorem4 y−20=180Combine like terms4 y=200Var one side∧¿other side y=50

118) Each table represents a line on the graph. On the lines below each table, identify the line that matches with it.

119) Identify the general rule for the following sequence: 1, 4, 7, 10… and list the 6th and 10th term.

120) The owner of an arcade measured the outside temperature each Saturday at noon and recorded the number of people in the arcade. The scatter plot shows the data collected. Which description fits the data in the scatter plot?

a. As temperature increases, more people come to the arcade.

b. As temperature increases, fewer people come to the arcade.

c. The number of people at the arcade is always the same.

d. There is no relationship between temperature and the number of people at the arcade.

121) The table shows the prices of stainless steel pots.a.Make a scatter plot of the data.b.Draw a line of best fit on the graph provided below.c.Write an equation of the line of best fit.

y=52x+0

d.Predict the cost of a 32-quart stainless steel pot.

y=52168

1y=40

e.What does the slope represent in your equation?For every 2 quarts increase the cost increases $5

122) Antonio and Juan are in a 4-mile bike race. The graph shows the distance of each racer (in miles) as a function of time (in minutes).

W F N

y=3 ( x−1 )+1y=3 x−3+1y=3 x−2

y es a

no

no

no

y=208x+0

y=104x+0

y=52x+0

x y1 12 43 74 106 1610 28

Who wins the race? ______________________

How do you know? _______________________

Solve the following equations. Please be sure to show ALL your work!

123)

124)

125)

126)

127) Evaluate: 3h + s if .

128) Convert the following to slope-intercept form

129) Write a linear equation in slope-intercept from the given points: (0, 3) (5, 0)

Slope= (3 )−(0 )(0 )−(5 )

Slope= 3−5

130) You are draining your fish aquarium. After 2 minutes, there are 6 gallons of water in the aquarium. After 5 minutes, the aquarium is empty. Write an equation that represents the volume y (in gallons) of water in the aquarium after x minutes.

Slope= (6 )−(0 )(2 )−(5 )

Slope= 6−3

Slope=−2

131) The table shows the value y (in dollars) of a car after x years of ownership. (The value of a car is how much the car is worth.)

a) Write a linear function in slope-intercept form that relates the value of the car(y) to years(x).

Slope=(24,000 )−(20,000 )

(0 )−(1 )

Slope=−4,000

b) What does the slope represent in your equation?The car depreciates $4000 every year. (Its value goes down this much every year).

c) What does the y-intercept represent in your equation?The car value when it was purchased.

d) What is the value of the car after 2.5 years?

e) After how many years will the car be worth nothing ($0)?

132) Write the equation of the line shown in the graph.

y=−2 ( x−2 )+6y=−2x+4+6y=−2x+10

JuanWins :Time is 13min .Antonio ' s : time is15min .which is2min . slower

y=54

x=54

x=−7

Infinite Solutions

x=196

∨x=3 16

y=−25x+3

y=−4000 x+24000

y=−35

( x−0 )+3

y=−35x+3

y=(−4,0001 )(52 )+24,000y=−10,000+24,000

y=$14,000

0=−4,000 x+24,000−24,000=−4,000 x

x=6 years

Slope= (5 )−(1 )(1 )−(3 )

Slope= 4−2

Slope=−2y=−2 ( x−1 )+5y=−2x+2+5

(1 ,5)

(3,1)

29113

y

12794 xx

2 x 3 4 2 x 1

155

3

xx

323

61,5 sandhp

2x 5y 15

133) Graph the following: −2 x+8 y=−16

134) Graph the following: y=−23x+5

135) Graph the following: y>−5 x+7

136) Which is steeper, a slide that rises 3 feet for every 2 feet of run, or a sliding pole that rises 5 feet for every 3 feet of run? Justify your answer.

137) Data from 200 flights was collected. The flights were categorized as domestic or international and late or not late.

a. Find the relative frequency (percent) of a flight being late.

b. Find the relative frequency (percent) of a flight being late among international flights.

c. Compare the relative frequencies. Circle one. International flights are less likely / equally likely / more likely to be late than flights in general.

138) Find the value of x.

139) You are pumping gasoline at a convenience store. The tank can hold at most 16 gallons. The function y = 3x represents the cost y (in dollars) of x gallons of gasoline.

a) Yes or no: Is the number 30 in the domain? Explain.______________________________

b) Yes or no: Is the number 30 in the range? Explain.______________________________

140) Solve the following systems. Coordinate planes have been provided if you choose to solve by graphing. However, you may solve using any method.

a)

Yes ,y = $cost, which is between $0- $48

m1=32

m1=112

m2=53

m2=123

1 23>1 12

m2>m1

40200

= 20100

=20%

1050

= x100

=20%

x y

0 -2

8 0

4 -1

4 -1

30150

=15=20%

40200

=15=20%

60+3 x+x+20=1804 x+80=1804 x=100x=25

No ,x = gallons tank can hold = 16 gal.

Substitution4 (8+3 y )+5 y=−232+12 y+5 y=−232+17 y=−217 y=−34y=−2

x=8+3(−2)x=8−6x=2

Solution = (2, -2)

1050

=15=20%

x 83y4x 5y 2

Late Not Late TOTALDomestic 30 120 150International 10 40 50TOTAL 40 160 200

b)Substitution4 x−7=−x+85 x=15x=3

y=4 (3 )−7y=12−7y=5

Solution = (3, 5)

141) A diver 20 feet below the surface of the water starts to ascend at 1/2 foot per second. A turtle that started 11 feet below the surface descends at a rate of 1/4 foot per second. Write an equation to model the position of each. When are they at the same position? What is that position?Substitution12x−20=−1

4x−11

2x−80=−x−444

3 x=36x=12 sec .

142) 1/1997 + 2/1997 + 3/1997 + … 1996/1997 =

143) x% of x% = x%

144) Simplify using logarithms: 661869

yd=12x−20

y t=−14x−11y=12(12)−20

y=6−20y=−1414 feet beneath the water

998Solution = {0, 100}

log x=log661869log x=1869 log 66 x=103400.727616

¿100.727616103400

¿100.727616103400

¿5.3409 x103400

y x 8y 4x 7