unit 1: lessons 1 and 2 - duke tip · unit 1: lessons 1 and 2 unit 1 exam solutions ... lessons 1...

Unit 1: Lessons 1 and 2

Unit 1 Exam solutions

Algebra 1

© 2009 Duke University Talent Identification Program

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Show all of your work in order to receive full credit.

1. Evaluate the expression below for 4x = − and 2y = :

3 23 2x y

xy

−

( ) ( )

( ) ( )

( ) ( )

( )

3 23 4 2 2 3 64 2 4

4 2 8

192 8

8

200

8

25

− − − −=

− −

− −=

−

−=

−

=

2. Evaluate:

( )34 10 2 16

10 4

÷ − + −

( ) ( )34 10 2 16 64 8 16

10 4 10 4

8 16

6

24

6

4

÷ − + ÷ + =

− −

+=

=

=



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3. Write an inequality to represent the situation below. Label your variables clearly. Solve and interpret your results.

How many $8 compact discs can Harry buy with the $42 his grandmother gave him for his birthday?

C = number of CDs Harry can buy 8 42C⋅ ≤

8 42

8 42

8 8

5.25

C

C

C

⋅ ≤

⋅≤

≤

Since Harry can’t buy a fractional part of a CD, the greatest

number of CDs he can buy is 5.

Answer: 5 CDs

4. Simplify the following:

a) ( )4 6 5 2x x− +

( )4 6 5 2 4 30 12

8 30

x x x x

x

− + = − −

= − −

b) ( )5 3 5 2 7j k j+ − +

( )5 3 5 2 7 15 25 2 7

13 25 7

j k j j jk j

j jk

+ − + = + − +

= + +



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5. Evaluate the following:

a) 4 6 18

5 7 70

− +

4 6 18 24 18

5 7 70 35 70

48 18

70 70

30

70

3

7

− − + = +

−= +

−=

= −

b) ( )8 7 2 5

3 8

− +

− −

( )8 7 2 5 8 14 5

3 8 11

6 5

11

6 5

11

11

11

1

− + − +=

− − −

− +=

−

+=

−

= −

= −

6. Evaluate ( ) 25 2 3x x x− + if 2x = −

( )( ) ( ) ( )

( )

22 5 2 2 3 2 2 10 2 3 4

2 12 12

24 12

12

− − − + − = ⋅ − − + ⋅

= ⋅ − +

= − +

= −



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7. True or False: ( )229 9− = − . Explain your answer.

FALSE: To evaluate 29− , we square first and then multiply by -1 to

get -81. To evaluate ( )2

9− , we multiply -9 times -9 to get 81.

Therefore, ( )229 9− ≠ −

8. Define variables, write an equation, and solve the following problem:

Alex drove 225 miles in 4 hours. What was his average rate in

miles per hour?

D = distance traveled; r = average rate; t = time traveled.

( )225 4

225 4

4 4

56.25

D r t

r

r

r

= ⋅

= ⋅

=

=

Alex’s average rate was 56.25 mph.

9. Solve for a. Round answer to nearest hundredth.

( )5.27 3.5 4.71 2a a+ = −

( )5.27 3.5 4.71 2

5.27 3.5 9.42 4.71

4.71 4.71

9.98 3.5 4.71

9.98 5.92

9.98 9.98

0.59

a a

a a

a a

a

a

a

+ = −

+ = −

+ +

+ =

=

=



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10. Solve the following equations for x. Leave non-integer solutions as fractions.

a) 2 3 29x − =

2 3 29

2 32

16

x

x

x

− =

=

=

b) ( )23 5 8

7x − =

( )

( )

23 5 8

7

7 2 73 5 8

2 7 2

3 5 28

3 33

11

x

x

x

x

x

− =

⋅ − = ⋅

− =

=

=

c) ( ) ( )2 4 5 3 4 5 12x x+ = − +

( ) ( )2 4 5 3 4 5 12

8 10 12 15 12

8 10 15 24

23 14

14

23

x x

x x

x x

x

x

+ = − +

+ = − +

+ = − +

=

=

d) 4 3 7 5x y z− = +

4 3 7 5

3 7 5 4

3 7 5 4

3 3

7 5 4 7 5 4 or

3 3

x y z

x y z

x y z

y z y zx x

− = +

− = + −

− + −=

− −

+ − − − += =

−



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11. Solve the following inequalities, and graph their solutions on a number line.

a) ( )2 3 6 8 2x x− ≤ − +

( )2 3 6 8 2

6 12 8 2

14 14

1

x x

x x

x

x

− ≤ − +

− ≤ − +

≤

≤

b) 4 5 3 23x− < − ≤

4 5 3 23

4 5 3 and 5 3 23

9 3 and 3 18

3 and 6

3 6

x

x x

x x

x x

x

− < − ≤

− < − − ≤

− < − − ≤

> ≥ −

> ≥ −

0 1

0 -3 -6 3 6 9 12 -9 -12



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12. Jackie wants to rope off a rectangular plot of land for her vegetable garden. She only has 90 feet of rope, and she wants the garden to be

twice as long as it is wide. What should the dimensions of her garden be?

w = width of plot

2w = length of plot

( ) ( )2 2 2P w w= +

( ) ( )2 2 2

90 4 2

90 6

15

P w w

w w

w

w

= +

= +

=

=

Dimensions: width = 15 ft; length = 30 ft.

w

2w

unit 1: lessons 1 and 2 - duke tip · unit 1: lessons 1 and 2 unit 1 exam solutions ... lessons 1...

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