Q1 Exam Review
1. Solve for x.
(x  5)(x  4)  (x  2)(x  10 )
[x=0]
2. Solve for x.
(x  6)( x  4)  20  x(x  2)  4
[ x can be any real number ]
2
3. Solve this equation for c: b  7bc  2c  6a  0 .
[
[ a: y = (3/4)x – 3; b: y = (-4/5)x + 1; c: y = 4/3 ]
4. Solve each equation for y.
a.
b.
c.
]
3x  4 y  12
2(4 x  5y)  10
(y  4)(y  2)  y(y  3)  4
5. Without graphing, find the equation of the line that passes through the points (– 4, 3) and (6, –2). Show all
work in a clear and organized manner.
[ y   12 x  1 ]
6. Show and explain how to find the equation of the line that passes through the points (– 4, 8) and (5, –1).
[ y  x  4 ]
7. Consider the line containing the points (–1, 3) and (4, 6).
a.
Find the slope of the line.
b.
Find the y-intercept.
c.
Find another point on the line that has integer coordinates. Write the coordinates of the point you found.
d.
Write an equation for the line.
3
[ a: 5 , b: 18
, c: Answers will vary; (– 6, 0) is one example, d: y  53 x  18
.]
5
5
8. Solve for p:
2 p1
3
 10  p  R
[ p = 29 - 3R ]
9. Solve each of these equations for x.
a.
2(x  5)  18
3x 2  7 x  12  (x  4 )(3x  2)
b.
10. Rewrite the fraction
4 3
7 2
[ a: x = 14; b: x = -20/3 ]
using positive exponents.
[
11. Simplify the following. Show your work. 3  8  2  6  3  12
12. Simplify: a.
m3  m4
b.
72
43
]
[ -36 ]
h8
h2
[ a: m 7 , b: h 6 ]
13. Which of the expressions below are equivalent to
a.
4x 2
b.
d.
12 x 5
3x 3
e.
1
2 x 2
2
1
2 x2
 
1
2 x 2
? Make sure you find all of the correct answers.
c.
16x 4
f.
2x 2 2x 2
[ (a), (c), and (d) are equivalent. ]
14. Rewrite the expressions below with no negative exponents and then simplify. Show all of your work.
a.
m 0 m1m 2
m 1
b.
b(2b 3 )2
c.
t
 
t 3
t 3
[ a: m4 , b:
1
4b 5
, c: t ]
15. For each of the following, state how many solutions the equation has. [ a: 2; b: 1; c: none; d: 2 ]
a.
|2x – 3| = 16
b.
|–7x + 4| = 0
c.
|8 – x| = –2
d.
– |15 – 3x| = –9
16. Evaluate the expressions below for the given value(s).
[ a: -95; b: undefined; c: 2; d: 11 ]
17. A student simplified each of the following using the order of operations. She did one of the four incorrectly.
Which one? Once you have identified the incorrect simplification, help the student out by offering some
feedback on what she might have done wrong and how to do the problem correctly.
[ Part c is incorrect because the student did not do the addition and subtraction, left to right. Should
have gotten 34. ]
18. Simplify: [ a:
a.
2
3
a 5 b 3 , b:
ab 3a 4 
19. Simplify:
A.
2
3
3x
4y
xy2
2
c: 4x 6 , d: 2w4 ]
b.
8 xy8
16 y6

c.
4x 3  x 3 =
f (3)
14w 6
7w 2
=
15 x 2 y
20 xy2
B.
3x 3 y 3
4
C.
9
16 y
b.
f (2)
x
5y
D.
20. If f (x)  4x 2  5x  2 , find the following:
a.
d.
c.
f

[ a: – 49, b: – 4, c: – 1.5]
1
2
21. Is x = 3 the solution to the equation 8  3(x  1)  5x  1 ? How do you know? Justify completely.
[ Yes, it is and students can justify by substituting 3 in for x and showing that each side is equal to the
same amount. ]
22. For each situation below, write the equation of the situation described.
a.
A line through the points (–12, 3) and (8, 15).
b.
A line with a slope of 4 and a y-intercept of 0.4.
c.
For her birthday, Louise got $100 from her Grammy, but she has been spending $10.00 each week.
d.
A line has intercepts of (-6, 0) and (0, -12).
[ a: y = (3/5)x + (51/5); b: y = 4x + .4; c: y = –10x + 100; d: y = –2x – 12 ]
23. Sketch each of the following on your paper. Write the areas as a product and a sum.
a. 3(2x  1)
b. x(x  2y  1)
c. (x  4)(2x  2)
d. (2y  1)(3x  y  2)
[ a: 3(2x  1)  6x  3 , b: (x)(x  2y  1)  x 2  2xy  x , c: (x  4)(2x  2)  2x 2  10x  8 ,
d: (2y  1)(3x  y  2)  6xy  2y2  5y  3x  2 ]
24.
A group of four students had a disagreement about the expression 2x 2  2x . Rashad thinks it simplifies to
4x 2 . Houston thinks it simplifies to 4x . Marianna thinks it simplifies to 2x 3 . Lakisha thinks it’s
already simplified. Settle their disagreement by writing a convincing argument for or against their
answers. [ Lakisha is correct. ]
25. Rewrite each of the following expressions using either a generic rectangle or the Distributive Property.
a. (3  x)(2x  1)
b. (2x  y)(x  2y)
c. 4y(2x  5)
d. (3e  2)(2x  y  4)
[ a: 2x 2  7x  3 , b: 2x 2  3xy  2y2 , c: 8xy  20y , d: 6ex  3ey  12e  4x  2y  8 ]
26.
Study the figures in the pattern below.
a. Draw Figures 4 and 5 on graph paper.
b. How many tiles will Figure 13 have? Show how you found your answer.
c. How tall will Figure 106 be? Be clear enough so that a new student to the class would understand.
[ a: See solution at right; b: 41 tiles; c: Since the height of each figure is 2
more than the figure number, the height of Figure 106 will be 108. ]
Figure 4
27.
Solve the diamond problems below.
a.
pro duct
#
#
b.
3
Figure 5
c.
-14
-1
6
sum
-5
4
[ a: x  3 , x  y  4 ; b: x and y are 2 and –7; c: y  2 , x y  1 2 ]
28. Write an algebraic expression representing the
tiles shown at right.
x
x2
[ 2x 2  3x  3  1 or 2x2  3x  2 ]
x
x
collection of algebra
= –1
x2
29. Is it true that (x  3)2  x 2  9 ? Justify your answer.
[ No; explanations will vary. ]
30. Solve each of the following equations for x and check your work.
a. 1 (x  2)  3x  7
= +1
b.
x2
1

[ a: x  4 , b: x  3 ]
x1
2
31. Find each of the following products by drawing and labeling a generic rectangle or by using the Distributive
Property.
a. (x  2)(x  5)
b. y(2y  2)
[ a: x 2  7x  10 , b: 2 y2  2 y ]
32.
Each point on the graph below represents a different person. Describe each person according to age and
amount of money he/she has saved.
[ Answers will vary. A is the youngest and has the least amount saved. B is
older than A and younger than C and has the most money saved. C is the
oldest and has an amount in savings that lies between A and C. ]
B
C
Money
Saved
A
Age
33.
Complete each of these Diamond Problems:
a.
b.
c.
[ a: x y  1 0, x  y  3 ; b: y  2 , x  y  2 ; c: x and y are –2 and –8, in no particular order. ]
Each point on the graph at right represents a different kind of boat.
a.
b.
c.
Which two boats cost the same amount? Explain how you found
your answer.
B
price
34.
C
A
D
Kara analyzed the graph and decided that boat D could carry the
greatest number of people. Do you agree with her conclusion?
Explain.
passenger capacity
Add a point to the graph for a boat (E) that costs more than boat C
but carries the same number of passengers that boat C carries.
[ a: Boats B and C cost the same amount since they have the same y-value. b: Yes, Kara is
correct. Point D is plotted to the far right of the x-axis indicating the highest passenger
capacity of the four boats. c: Point E should lie directly above point C. ]
35. Find the missing dimensions (length and width) or area of each part and write the area of the rectangle as a
–1
-20
5xy
x
+1
x
5y
–4
product and area.
[ a: –1x, 5y, 100y, 20, ; (1  5y)(x  20)  5xy  1x  100y  20 , b: 5xy  5y , 4x  4 ;
(x  1)(5y  4)  5xy  4x  5y  4 ]
36. Each of the following expressions represents the area of a rectangle and are written as either a product or a
sum. Decide as a group which form each is currently in (product or sum) then rewrite it as a product and a sum.
b. 10x  20 
a. 4(x  5) 
c. x 2  3x
d. 3(2y  5) 
4(x  5)  4x  20 , b: 10x  20  5(2x  4) or 10(x  2) or 2(5x  10) , c: x 2  3x  x(x  3) , d:
3(2y  5)  6y  15 ]
[ a:
37. Write the area of the rectangle below as a product and as a sum. [ (x  2)(x  y  3)  x 2  5x  xy  2y  6 ]
1
1
x
x
y
11 1
38. Write the letters of the steps in the order that will solve 2x  4y  10 for y. You may not need to use all
four steps listed.
[ B, A or B, C, A are possible answers ]
A. Divide both sides of the equation by – 4.
B. Subtract 2x from both sides of the equation.
C. Add 10 to both sides of the equation.
D. Divide both sides of the equation by 2.
39. Build rectangles with the dimensions given below. Sketch each rectangle on your paper, label its dimensions, and
write an equivalence statement for the area as a product and a sum.
a.
(2x  2)(x)
b.
(x  1)(x  y  1)
[ a: (2x  2)(x)  2x 2  2x , b: (x  1)(x  y  1)  x 2  xy  2x  y  1 ]
40. Consider the equation mat at right.
a. Write the equation algebraically.
b. Solve the equation for x. Show all your steps.
c. Check that your solution is correct. Be sure to show how you
checked your solution.
+
x
x
x
x
x
–
x
x
x
–
[ a: 4x  5  2(x  1)  3x  3(x  1) ; b: x  2 ]
41. Consider the rule y  x 2  3 .
a.
Make a table and a complete graph for the rule. Be sure to discuss proper scaling with your
team before you begin.
b.
What does your graph look like?
[ a parabola opening upward ]
42. Solve the following equations for the indicated variable. Show all of your work.
a. Solve for x: 2(x  1)  x  12
b. Solve for y: 8x  2y  4
c. Solve for m: 4 p  4  2(m  p)
d. Solve for x: y  4x  3
e. Solve for x: 2(5x  10)  50
f. Solve for y: x  8y  24
g. Solve for w: 4(4  t)  1 w  4
h. Solve for x: 2x(x  3)  (2x  1)(x  4)
[ a:
x  10 ,
b:
y  4x  2 ,
c:
m  3p  2 ,
d:
x
1
4
y
3 , e:
4
43. Find the area of each rectangle below.
a.
50
44.
f:
y   18 x  3 ,
g:
w  19  4t ,
h:
x4]
[ a: 195 square units, b: 95 square units ]
b.
40
40
x  7 ,
12
7
14
36
For two tile patterns on their resource page, Janice and Karen made the x  y tables shown below.
a.
Describe any similarities and differences you can see in the two tables.
b.
Graph each pattern on a single set of axes.
c.
Explain any connections you see between the graphs and the tables.
x
y
x
y
Figure #
# of Tiles in Pattern 1
Figure #
# of Tiles in Pattern 2
0
1
0
1
1
3
1
5
2
5
2
9
3
7
3
13
[ Pattern 1 has a growth of 2 . Pattern 2 has a growth of 4. Both patterns have a single tile for Figure 0, and both graphs cross the y-axis at the point (1, 0). ]
45.
Find the mistake in the solution below and explain how to correct it. Find the correct answer.
Algebra
3x  5  3x  x  8  x 1
6x  5  2x  7
4x  5  7
4x  12
x3
Steps
original problem
combine like terms
subtract 2x from both sides
add 5 to both sides
divide both sides by 4
[ Like terms were not combined correctly in the second step (x – x = 0). The right side of the equation should have the 7 without the 2x; x = 2. ]
46.
Mariela graphed all of the equations below but forgot which equation went with which graph. Help her
match each equation with the appropriate graph. Discuss the answers with your group and write a few
sentences explaining how you figured it out.
b y
c
y x3
d
y  2x  1
y
1
2
x
x 1
a
y  2x  3
y  12 x  1 . Line (b) is y  2x  1 .
y  2x  3 . Line (d) is y  x  3 .]
[ Line (a) is
Line (c) is
47.
Study the tile pattern at right.
Figure 0
Figure 1
a.
Draw Figure 3 and Figure 4. Explain how the pattern grows.
b.
Write an equation (rule) for the number of tiles in the pattern.
c.
Explain how the growth factor appears in your equation.
[ a: Each new figure has an additional tile on each “arm,” and the length of each arm matches the figure number, b:
coefficient (3) shows the number of new tiles added to each new figure. ]
48.
Write a rule for each graph below.
y
a.
b.
•
4
3•
-3
10
•
[ a: y  3x  3 , b: y  4x  10 ]
3
x
1
y  3x  1 ,
c: The