Lesson 2: Solving Equations and Inequalities
Solve It! Quiz solutions
Show all of your work in order to receive full credit.
1. Tashelle solved the problem
2
3
( x − 11) + 3 = 51 ( x + 4)
and determined that
x = 17 . 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.
( x − 11) + 3 = 15 ( x + 4 )
10(x − 11) + 3 = 3 ( x + 4 )
2
3
10 x − 110 + 3
10 x − 107
7x
x
= 3x + 12
= 3x + 12
= 119
= 17
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:
( x − 11) + 3 = 15 ( x + 4 )
10(x − 11) + 45 = 3 ( x + 4 )
2
3
10 x − 110 + 45
10 x − 65
7x
x
Algebra 1
© 2009 Duke University Talent Identification Program
Page 1 of 3
= 3x + 12
= 3x + 12
= 77
= 11
Lesson 2: Solving Equations and Inequalities
Solve It! Quiz solutions
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.
Stairs
Tub
Porch
House
a) Given the constraints above, determine the maximum length of the
porch.
If the length of the porch is x, then the width is
side with the stairs measures
1
3
x . The
1
3
x − 3 (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
( 13 x ) + ( x ) + ( 13 x − 3) + (5) + (7 − 2 ) . Since Mr. Johnston has
45 feet of fencing, we set the expression above equal to 45
and solve for x.
( 13 x ) + ( x ) + ( 13 x − 3) + (5) + (7 − 2) = 45
1
3
x+x+
1
3
x − 3 + 5 + 5 = 45
5
3
x + 7 = 45
5
3
x = 38
x = 38 ( 35 )
x = 22.8
The maximum length of the porch is 22.8 ft.
Algebra 1
© 2009 Duke University Talent Identification Program
Page 2 of 3
Lesson 2: Solving Equations and Inequalities
Solve It! Quiz solutions
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.
Algebra 1
© 2009 Duke University Talent Identification Program
Page 3 of 3