Geometry

Pre-Requisite Skills

Packet

Section 1-3 page 16

Real Numbers and the Number Line

A number that is the product of some other number with itself, or a number to the second power, such as 9 = 3 × 3 = 32, is called a

perfect square. The number that is raised to the second power is called the square root of the product. In this case, 3 is the square root

of 9. This is written in symbols as Sometimes square roots are whole numbers, but in other cases, they can be estimated.

What is an estimate for the square root of 150?

There is no whole number that can be multiplied by itself to give the product of 150.

10× 10 = 100 11× 11 = 121 12× 12 = 144 13× 13 = 169

You cannot find the exact value of 150 , but you can estimate it by comparing

150 to perfect squares that are close to 150.

150 is between 144 and 169, so 150 is between 144 and 169 .

144 150 169

12 150 13

The square root of 150 is between 12 and 13. Because 150 is closer to 144 than it is to 169, we can estimate that the square root of 150

is slightly greater than 12.

Exercises

Find the square root of each number. If the number is not a perfect square, estimate the square root to the

nearest integer.

1. 4 2. 50 3. 25

4. 81 5. 121 6. 10

7. 15 8. 225 9. 64

9

10.

25

81

11.

225

169

12.

1

625

13. 0.64 14. 0.81 15. 6.25

Section 1-3 page 16

Section 1-2 page 10

Order of Operations (continued)

Simplify each expression.

7. 42 8. 53 9. 116

10.

25

6

11. (1 + 3)2 12. (0.1)3

13. 5 + 3(2) 14.16

4(5)2

15. 44(5) + 3(11)

16. 17(2) 42 17.

320 2

10(3)5

18.

327 12

8 3

19. (4(5))3 20. 25 42 22 21.

43(6)

17 5

Section 1-7 page 47

Distributive Property (continued)

Simplify each expression.

10. – (14 + x) 11. – (-8 – 6t) 12. – (6 + d)

13. – (-r + 1) 14. – (4m – 6n) 15. – (5.8a + 4.2b)

16. – (-x + y – 1) 17. – (f + 3g – 7) 18. -4(3x – 2y + 4)

Section 1-5 page 30

Section 1-6 page 38

Section 1-7 page 47

Section 1-2 page 10

SOLVING EQUATIONS

Solve for the variable. Show all the steps. All answers will not be integers; write these as a mixed number. 1. 1512 =+x 2. 25411 =+− x 3. 22513 =− x 4. 1037 −=− x 5. 2816 =+− x 6. 027 =+x

Objective: To solve equations using more than one transformation.

Example 1 Solve for x. 3x – 7 = 8 3x – 7 + 7 = 8 + 7 Add 7 to both sides of the equation 3x = 15 Combine like terms x = 5 Divide both sides by 3

1

Solve for the variable. Show all the steps. All answers will not be integers; write these as a mixed number. 7. 9152 =+− xx 8. xx +=− 120907 9. 188 +=+ xx 10. 18040554 =+++ xx 11. xxx 18281322 −−−=− 12. 171852 +−=++ xx

Example 2 Solve for x 12 + 3x = 2 + 5x 12 + 3x – 3x = 2 + 5x – 3x Subtract 3x from both sides 12 = 2 + 2x Combine like terms 12 – 2 = 2 – 2 + 2x Subtract 2 from both sides 10 = 2x Combine like terms 5 = x Divide both sides by 2

2

Multiplying with Monomials

. (3q3b) (502b) = 3 .503+21:)1+1 =15 0 5 b2

(_~q~O)4 =H~)4 (d~)4 (b)4 =-2 • -2 • ;2· -2d204b H = 1608b 4

.: . >'

1. (4C)2

8. X2 (-2xz) (4Z5)

4.2x(-xy)(_y2) 10. (-x) (-2xy) (-3xyz)

6. 3s (-2st)2

3 Q. 0

. .. .. __.._._---­

DISTRIBUTIVE PROPERTY

Apply the distributive property. 1. )2(4 2 xx + 2. )7(2 2 xxx −

3. )32(8 2 +− xxx 4. 25 ( 17)x x −

5. 2 24 ( 3 12)x x x− + − 6. 3 2(4 16 27)x x x− − −

7. 4 24 ( 25)x x− − 8. 210 (3 6 8)x x x− + −

Example Distribute )32(2 22 −+ xxx = )32(2 22 −+ xxx Use distributive property = )32()22()2( 2222 •−•+• xxxxx Distribute 22x to every term in parenthesis = 234 642 xxx −+ Multiply and write answer in decreasing powers of x

Objective: To multiply a variety of polynomials using the distributive property.

3b

SOLVING EQUATIONS USING THE DISTRIBUTIVE PROPERTY

Solve for the variable. Show all the steps. All answers will not be integers; write these as a mixed number. 1. 6)3(2 =+− x 2. 0)7(5 =+x 3. 115)4(6 =+−x 4. 11)2(73 +−=− x 5. 8)514(2 −=+− x 6. 64)7( =−+− x

Objective: To solve equations involving the distributive property.

Example 1 Solve for x. 3(4x – 5) = 9 12x – 15 = 9 Distribute the 3 12x – 15 + 15 = 9 + 15 Add 15 to both sides 12x = 24 Combine like terms x = 2 Divide both sides by 2

4

Example 2 Solve for x.

12)6(21

=+x

12321

=+x Distribute

3123321

−=−+x Subtract 3

921

=x Combine like terms

2)9()21(2 •=• x Multiply by reciprocal

18=x Simplify Solve for the variable. Show all the steps. All answers will not be integers; write these as a mixed number.

7. ( ) 11531

−=−x 8. ( ) 181052

=+x 9. ( ) 31243

−=+− x

5

Solve for the variable. Show all the steps. All answers will not be integers; write these as a mixed number.

10. ( ) 112

6=

−x 11. ( ) 86

4=

−−

x 12. ( ) 25

7−=

+−

x

Example 3 Solve for x.

34

3=⎟

⎠⎞

⎜⎝⎛ +x

434

34 •=⎟⎠⎞

⎜⎝⎛ +x Multiply both sides by denominator to clear the fraction

123 =+x Simplify 9=x Subtract 3 from both sides

6

MULTIPLYING BINOMIALS

Use the FOIL method to multiply. Show ALL steps. 1. )2)(4( ++ xx 2. ( 7)( 8)x x+ −

3. ( 5)( 9)x x− + 4. ( 3)( 12)x x− −

5. (2 6)( 4)x x+ − 6. (3 5)(6 2)x x− +

7. ( 11)( 11)x x+ − 8. ( 15)( 15)x x+ +

Example Multiply )3)(2( −+ xx = )3)(2( −+ xx F (multiply the first terms together) = )3)(2( −+ xx O (multiply the outside terms together) = )3)(2( −+ xx I (multiply the inside terms together) = )3)(2( −+ xx L (multiply the last terms together) = 6232 −+− xxx Perform multiplication = 62 −− xx Combine Like terms

Objective: To multiply binomials using the FOIL method.

7

MULTIPLYING POLYNOMIALS

Multiply the polynomials. Show ALL steps. 1. )53)(4( 2 +++ xxx 2. 2( 5)(2 8)x x x− − − 3. 2(2 4)( 3 2)x x x− + + 4. 2 2( 3)( 6 12)x x x+ + + 5. )73)(62( 22 +−− xxx 6. 2(3 5)(6 2 3)x x x− + − 7. 3 2( 11)(2 3 10)x x x+ + − 8. 3 2( 15)(2 3 4)x x x+ + −

Example Multiply )34)(2( 2 −++ xxx = )34)(2( 2 −++ xxx Distribute the x = )34)(2( 2 −++ xxx Distribute the 2 = )682()34( 223 −++−+ xxxxx Perform multiplication = 656 23 −++ xxx Combine like terms

Objective: To multiply binomials and trinomials.

8

SL.OPE AND THE COORDINATE PL.ANE

Objective: To determine slope both algebraically and graphically. Slope = vertical change horizontal changl

Example 1 change in y-values Y2 - y,SJ ope = = -"--"'­

Calculate the slope of the line containing points: I change in x-values - x,x2

(2,4) and (3,7) (f." '/. ('4 y, ') (- L 5) (3, -2)

change iny change in x Formula to find slope slope = -2-5 = -z

3-(-1) 4 4-7

Substitute x and y values 2-3

-3 Simplify

-1

3 Simplify

Calculate the slope of the line containing the pairs of pOints. You may leave an answer as an improper fraction, but it should be in lowest terms.

1. (3,8) and (0,4) 2. (-1,2) and (6,5)

3. (-4, -1) and (9,-3) 4. (4,2) and (-7, -8)

9

Example 2

Use the graph to: a.) determine the slope of the line pictured on the coordinate plane b.) determine the point where the graph crosses the y-axis c.) write the equation of the line in slope-intercept form.

Choose two ordered pairs that are points on the line: (-1,2) and (2,9) a.) Use the method outlined in Example 1 to calculate the slope.

21

92−−− =

37

−− =

37

b.) The line crosses the y-axis at the point (0,4).

c.) The equation of the line is y = 37 x + 4.

Remember the slope-intercept equation: y mx b= +

10

Calculate the slope and determine the point where the graph crosses the y-axis. Write the equation of the line in slope-intercept form. 1. Show work here:

2. Show work here:

11

GRAPHING LINEAR EQUATIONS Example

The equation of a line is 23 −= xy . Identify the slope, y-intercept, and graph the equation. y-intercept is (0,-2)

Objective: Graph a line given the equation.

Slope is 3, which can

be written as 13

(change in y is +3, change in x is +1)

12

Use the coordinate planes to graph the following equations. 1. 3+−= xy 2. 52 −−= xy

3. 132

=+ xy 4. 027

=− xy

13

FACTORING GCF

Factor out the GCF in each polynomial. 1. xx 243 2 − 2. 32164 2 −+ xx 3. 23 23 xx − 4. 37 5025 xx − 5. xxx 601040 23 −+ 6. 22 1263 xyyxy +−

7. xxx 12114333 23 −+ 8. xyyxyx 961624 322 −+

9. 90225 2 −x 10. 352 84723660 yyyy −+−

Example Factor 23 42 xx + )2(2 2 +xx Factor out the 2x² that all the terms have in common

Objective: Factor using the greatest common factor (GCF).

14

FACTORING POLYNOMIALS

Factor the polynomials. 1. 24112 +− xx 2. 54152 ++ xx 3. 16102 +− xx 4. 6322 −+ xx 5. 3652 −− xx 6. 121222 ++ xx 7. 252 −x 8. 1002 −x 9. 1442 −x 10. 125202 −+x 11. 273 2 ++ xx 12. 932 2 −− xx

Example Factor 16102 +− xx 1. Since the coefficient of the x² term is one, just think of factors of your last term: +16 that add up to the middle term: -10. 2. The factors of +16 that add up to -10 are -8 and -2. 3. Therefore: 16102 +− xx = )8)(2( −− xx So, the solution is: )8)(2( −− xx **Hint: to check your answer, simply use the foil method. If you come up with the trinomial that you started with, you are correct!

Objective: Factor trinomials and difference of two squares.

15

THE QUADRATIC FORMULA Objective: To find the solution to a quadratic equation using the quadratic formula. Example

The formula is: 2 4

2b b acx

a− ± −

=

Solve for x using the quadratic formula. 01522 =−+ xx 1, 2, 15a b c= = = − Identify a, b, and c

=x 12

151422 2

•−••−±− Substitute values into the formula for a, b, and c

2 4 602

x − ± +=

2 642

x − ±= Simplify

2 82

x − ±=

2 82

x − += and 2 8

2x − −= Split the equation to find both solutions

62

x = and 102

x −= Simplify and reduce the fractions

x = 3 and x = 5−

16

Use the quadratic formula to solve for x. Not all answers will be integers. Some may contain radicals. Write your answers on the lines below each equation. Show all steps. 1. 021102 =++ xx 2. 0752 2 =−− xx X = _____ & _______ X = ______ & ______ 3. 92 =− xx 4. 422 =+ xx X = ______ & ______ X = ______ & ______

17

SIMPLIFYING RADICALS

Simplify the radical.

1. 24 2. 200 3. 125

4. 48 5. 363 6. 240

Objective: To simplify radicals without the use of a calculator.

Example

Simplify the radical.

27 = 39• = 9 • 3 = 3 3

18

OPERATIONS WITH RADICALS Perform the indicated operation. 1. 7476 + 2. 117118 − 3. 665 −

Objective: To incorporate operations with radicals.

Example 1

Add 53 + 57 = 510 Add the outside terms and keep the same radical (the same process is used with subtraction) Example 2 Example 3

Multiply Divide

15453 • 5

210

= 15543 •• = 22 = 7512 Simplify the radical = 32512 • = 32512 • = 3512 •• = 360

19

4. 158157 + 5. 3

512 6. 2

722

7. 532 • 8. 20857 • 9. 5610 • 10. 9911• 11. 2326 • 12. 48332 •

20

Solving Problems

Set up and solve each equation.

The sum of twice a 2n + 21 '" 83 number and 21 Is 83. 2n + 21 -21 = 83-21 Find the number. 2n = 62

n = 31 Th~ number Is 31.

1. Twice a number, diminished by 17 is -3.Flnd the number.

2. Six times a number, increased by 3 is 27. Find the number.

3. Three times the difference of 5 minus a number Is 27. Find the number.

4. Karl's team score Is 39 points. This was one point less than twice Todd's team score. Find Todd's team score.

5. The length of a rectangle is 6 feet more than twice the width. If the length is 24 feet, what is the width?

6. Four-fifths of the third grade went on a trip to the zoo, If 64 children made the trip, how many children are in the third grade?

7. The price of a pack of gum today Is 63C, This is 3C more than three times the price 10 years ago. What was the price 10 years ago?

21

FRACTIONS

Objective: To add, subtract, multiply and divide fractions.

Example 1

a. Add 4 35 5+ b. Subtract

5 29 9−

521

57

534

53

54

==+

=+ 31

93

925

92

95

==−

=−

Example 2 Example 3 Example 4

Add 3 54 6+ Multiply

138

43• Divide

91

32÷

1210

129

65

43

+=+ 138

43•

19

32•

9 10 19 7 or 1

12 12 12+

= 136 6

Hints for adding and subtracting fractions: o Find a common denominator o Do not write as improper fractions o Add/subtract numerators only o Check your answer for an improper fraction o Reduce the answer if necessary

Hints for multiply fractions:

o Write mixed numbers as improper fractions o Multiply the numerators across o Multiply the denominators across o Write product as a mixed number o Reduce if necessary

Hints for dividing fractions:

o Write mixed numbers as improper fractions o Dividing by a number is the same as multiplying by the reciprocal o Follow the last four steps from the hints for multiplying as stated above

Change to a multiplication problem

28

Perform the indicated operation. Show ALL work.

1. 213 +

326 2.

532 +

7110

3. 929 +

835 4.

412

325 −

5. 1092

546 − 6.

745 8 −

7. 116 4 • 8.

95

32•

29

Perform the indicated operation. Show ALL work.

9. 83

513 • 10.

712

94•

11. 5318 ÷ 12.

32

149÷

13. 812

416 ÷

30

SYSTEMS OF EQUATIONS (SUBSTITUTION) Solve each system of equations using substitution. Show ALL work.

1. 1624

102=+=+yx

yx 2.

1533135=+=+

yxyx

Objective: Solve the system of equations by substitution.

Example Solve

1152

93−=+

=−yx

yx

yxyx

yx

=−+==−

9393

93 Solve one of the equations in terms of a variable

11)93(52

1152−=−+

−=+xx

yx Plug 3x-9 in for y in the other equation

23417

1145171145152

==

−=−−=−+

xxx

xx

Solve for x

9)2(3

93=−

=−y

yx Plug 2 in for x in one of the equations

6 9 3

yy− == −

Solve for y

(2,-3) Write your answer as an ordered pair

31

3. 1

134=+=+

yxyx

4. 1

62−=+−

=−yxyx

5. 224

36−=−

=−yx

yx 6.

yxyx41

732−=

=+

32