Lesson 2: Solving Equations and Inequalities Solve It! Quiz solutions

Algebra 1

© 2009 Duke University Talent Identification Program

Page 1 of 3

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

1. Tashelle solved the problem ( ) ( )2 13 5

11 3 4x x− + = + and determined that

17x = . However, when she checked her answer, it didn’t work in the original equation.

Find Tashelle’s mistake in her work below, and explain what she did

wrong. Then calculate the correct value for x.

( ) ( )

( )

2 13 5

11 3 4

10( 11) 3 3 4

10 110 3 3 12

10 107 3 12

7 119

17

x x

x x

x x

x x

x

x

− + = +

− + = +

− + = +

− = +

=

=

She multiplied both sides by the LCD, which is 15, but when she

multiplied the left side, she forgot to distribute the 15 to the +3.

The correct way to solve this is:

( ) ( )

( )

2 13 5

11 3 4

10( 11) 45 3 4

10 110 45 3 12

10 65 3 12

7 77

11

x x

x x

x x

x x

x

x

− + = +

− + = +

− + = +

− = +

=

=

Lesson 2: Solving Equations and Inequalities Solve It! Quiz solutions

Algebra 1

© 2009 Duke University Talent Identification Program

Page 2 of 3

2. Mr. Johnston has decided that he wants to put a porch on the back of his

house. He wants the width of the porch to be 1/3 of the length. He is

planning to have a 5 ft by 7 ft hot tub at the corner of the porch and

wants to enclose the porch and the tub with a railing. He needs to leave a

2-ft opening in the railing to put a gate going into the tub and also wants

to leave an opening for a 3-ft door at the stairs. (See picture below.) At

the local home improvement store, he found 45 feet of railing on

clearance.

a) Given the constraints above, determine the maximum length of the

porch.

If the length of the porch is x, then the width is 13x . The

side with the stairs measures 13

3x − (since the opening for

the stairs is 3 ft). In addition, Mr. Johnston needs 5 feet of

fence to enclose the left side of the tub and 7 2 5− = feet

of fence to enclose the house side of the tub.

The amount of fence needed in terms of x is

( ) ( ) ( ) ( ) ( )1 13 3

3 5 7 2x x x+ + − + + − . Since Mr. Johnston has

45 feet of fencing, we set the expression above equal to 45

and solve for x.

( ) ( ) ( ) ( ) ( )

( )

1 13 3

1 13 3

53

53

35

3 5 7 2 45

3 5 5 45

7 45

38

38

22.8

x x x

x x x

x

x

x

x

+ + − + + − =

+ + − + + =

+ =

=

=

=

The maximum length of the porch is 22.8 ft.

Tub

House

PorchStairs

Lesson 2: Solving Equations and Inequalities Solve It! Quiz solutions

Algebra 1

© 2009 Duke University Talent Identification Program

Page 3 of 3

b) Write a note to Mr. Johnston stating your recommendations for the

dimensions of the porch. Give reasons to justify your

recommendations.

Dear Mr. Johnston,

The maximum possible length for your porch would be

22.8 ft, and the width would have to be 7.6 ft. However,

since it is difficult to measure 8/10 ft and 6/10 ft, my

recommendation would be to make the porch 21 ft x 7 ft.

It will be easier to measure, and you will have 2 ft left

over. If desired, you could make the gate for the tub out of

the leftover fence.