Download Algebra 3 Spring 2011 Final Review

Algebra 3 Spring 2011 Final Review
Equations: Slope: m 
Name:______________________________________
y2  y1
, Slope-intercept form: y  mx  b , Point-slope form:
x2  x1
Quadratic Formula: x 
b  b2  4ac
, Discriminant:
2a
Unit 1
Determine the numbers that are found in each set:
a. Natural numbers:________________________
b. Whole numbers:_________________________
c. Integers:_______________________________
d. Rational numbers:________________________
e. Irrational numbers:_______________________
f. Real numbers:___________________________
Insert >, < or = to make the statement true.
16
a. 22 _____ 11 b.  6 _____ 6
c.
_____
2
Simplify the following.
y  y1  m( x  x1 )
b2  4ac
Simplify the following.
a. 6  2  2  2
5
b.
16  13  5  32
17  5
Perform the indicated operation of real numbers:
2
25
4  1
a.     
b.   
15
6
7  7
d. 
c. 4  ( 6)
16
8

27
45
Evaluate the following for x = 1, y = -4, and z = 2
a. |-2| = _____
b. |4| = ____
c. |-5.6| =____
a. x  y  2
Write each fraction in lowest terms.
a.
15
=_____
10
b.
32
=______
44
c.
56
=______
42
Perform the indicated operation. Write your answer
in lowest terms.
a.
16 21
 =______
9 8
b.
5 15

=______
11 44
3 10
 =______
5 3
d.
7 3
 =______
15 10
1 5
e. 3  =_______
4 6
1
3
f. 5  2 =______
3
4
3
g. 2 1 =_______
4
4
2
h.  2 =______
3
3
x2  y  x  z
z3  y
Unit 2
Simplify the following expressions.
a. 4 x  x 2  1
b. 2  x 2  3 x   5 x
c. 3  x  2  12  x 
e.
c.
b.

d. 4  x  3  3  x  2
 
1
1
4 x 2  2 x  8  15 x 2  9 x  3
2
3

Translate the following into algebraic expressions or
equations.
a. Subtract 7 x  2 from x  5 __________________
b. The difference of a number and five, divided by
seven. ________________________
c. The sum of a number and six subtracted from two
times a number decreased by one.
__________________
d. Five times a number less four is the same as twice
the sum of the number and three.
___________________
Solve for x:
a. 3x  9  18
c.
x 2

6 3
b. x  4x  7  x  3  8
d.   5x  1  7 x  3
e. 9 x  x  1  6( x  1)  7
f.
5  x  1
6
Unit 3
Graph the following using coordinate points. Find at
LEAST three points:
3x  6 y  12
 2x  3
Solve for x. Write the answer in interval notation and
graph.
a. x  5
Identify the intercepts of the following graphs:
b. 2  x  3
c. 3x  4  x
Formulas
a. Given A  l  w ; A  56, l  8 . Find w.
Find the intercepts for the following equations:
a. x  4 y  8
b. 2 x  3 y  6
b. Solve for w: V  lwh .
Graph the following equations by plotting intercepts:
c. Mike is trying to replace the carpet in his
bedroom. His room is rectangular and has a width of
14 feet and a length of 20 feet. How much carpet
will Mike need for his bedroom?
d. The normal body temperature for a human is 98.6
degrees Fahrenheit. Express this temperature in
9
terms of degrees Celsius. F    C  32
5
4 x  5 y  20
Find the slope of the following:
a. Line through (4, -2) and (2, 5)
b. Line through (2, 3) and (2, -4)
Solve the following word problems.
a. Greg gets paid $20 dollars an hour. If Greg’s
paycheck at the end of week is $700 before taxes,
how many hours did he work?
c. Line through (8, -1) and (3, -1)
Determine if the following lines are parallel,
perpendicular, or neither.
a. L1: (5, -1) and (4, 3); L2: (2, 3) and (-1, 15)
b. The sum of two consecutive even numbers is 178.
What are the two numbers?
b. L1: (3, -2) and (0, -1); L2: (2, 5) and (3, 2)
c.
y  3 x  5
6x  2 y  1
d.
2 y  5 x  4
4 x  10 y  20
Find the slope and y-intercept of the following:
5
a. y  x  1
b. 4 x  6 y  2
6
c. y  5
Unit 4
Determine if the following relations are functions or
not:
a. 1,3 , 1, 4  ,  3, 4 
b.  5,3 ,  2,3 ,  6,3
c.
d. x  4
5
-1
3
2
e.
d.
0
6
-4
f.
Graph the following using slope-intercept form:
a. y 
1
x 3
4
Find the domain and range of each function:
a.  2,3 ,  2,1 ,  0,1 ,  4, 5 
b. 3 y  2 x  6
b. f ( x)  5 x  1
c.
d.
Use point-slope form to find the following equations:
a. Find an equation of the line passing through (3, 4)
1
with slope . Put in slope-intercept form.
3
Determine if the following are solutions to the given
system of linear equations:
2 y  4 x
a. 
1) (-3, -6)
2) (0, 0)
2 x  y  0
b. Find an equation of the line through (4, 0) and (3,
-2). Write the equation in slope-intercept form.
2 x  y  1
b. 
3x  y  0
c. Find an equation of the line perpendicular to the
3
line y  x  4 and goes through the point (6, -1).
2
Write the equation in slope-intercept form.
1) (3, -5)
2) (-1, 3)
Solve the following systems by graphing. If a single
point, write as an ordered pair:
y  x  2
a. 
 y  5 x  4
Solve the following systems by substitution. If a
single point, write as an ordered pair:
1
4 x  2 y  5
 x y 3
a. 
b.  2
2 x  y  4
 x  6  2 y
Solve the following systems by elimination. If a
single point, write as an ordered pair:
3x  y  5
2 x  3 y  0
a. 
b. 
6 x  y  4
4 x  6 y  3
Unit 5
Determine the degree of the following polynomials.
Then indicate whether the polynomial is a
monomial, binomial, trinomial, or none of these.
a. 5x 2 y  4  x3 y 2
b. 16  5x3 y 2
c. x  y 3  xy  1
c.
4x 3 z 2
Perform the indicated operations and simplify.
a.  3 x5  2 x3  x 2    3x 2  x5  x3 
b.  4 xy  3  2 x 2 y    7 x 2 y  2 y 2  6 xy 
Problem solving with systems:
a. The sum of two numbers is 62. The first number
is six less than three times the second. What are the
two numbers?
c.
 3x  4   x 2  5 x  2 
d.
x
2
 7 x  6    x  1
Unit 6
b. The sum of two numbers is 84. The first number
is nine less than twice the second. What are the two
numbers?
Factor using Greatest Common Factor.
3
2
a. 4c  18c  8c
b. x 4 y 3 z  2 x 2 y 2 z 3  4 x3 y 2 z
Factor by grouping. Remember GCF!
a. x3  5 x 2  x  5
b. 3x 2  6 x  7 x  14
c. A jewelry maker spends $1650 on operating costs
and $35 for each necklace made. The necklaces are
then sold for $85 each. How many necklaces does
the jewelry maker need to sell in order to break even?
Factor the following trinomials. Remember GCF!
a. x 2  5 x  6
b. 2 x 2  8 x  6
c. 2 x 2  x  10
d. 6 x 2  15 x  36
Pre-Unit 5
Write in simplest form without zero or negative exponents.
a. 5t 4  3t 3
c.
 r st 
3
4
b.
3
x3 y 2 z
d.
xy 5 z 3
3
 x3 
f.  5 
y 
6 xy 2
e.
9 xy
 2a bc 
g. 
2 4 
 15a b c 
5
 2x y z  xy z 
2
0
2 3
Factor the following binomials (diff. of squares, sums/diff.
of cubes). Remember GCF!
2
a. x  1
2
b. 25 x  36
3
c. 8 x  27
3
d. 3x  24
e. 9 x 2  100
f. 125 x3  1
4
6 2
h.
2
ab
a 3b5
Solve by factoring. Remember GCF!
a. x 2  5 x  4  0
b. 3 x 2  8 x  5  0
c. 12 x3  27 x  0
d. x 2  6 x  7  0
Solve using completing the square.
a. x 2  8 x  12
b. 2 x 2  8 x  22
Solve using quadratic formula and determine the number
and types of roots using the discriminant.
a. x 2  6 x  7  0
b. 4 x 2  28 x  49  0
c.  x 2  x  1  0
d.  x 2  5 x  6  0
Unit 7
Simplify each rational expression.
x2  4
2
b. x  6 x  8
ax 2 y
a.
ax 2 z
x  5 x  24
x 2  7 x  30
x 4
d. x  9 x  14
2
c.
2
a.
2x 1 2x  7

3x
3
c.
4x2
1

2x 1 2x 1
5 x3 y y 3

x 2 y 2 15x 2
x2  4 x  1

x2 1 x2  2x
b.
12
x2  1
c. 2

x  x 4x  2
2 x 2  6 x 9 x  81

x 2  18 x  81 x 2  9
d.
Divide the rational expressions. Write your answer in simplest
form.
3 y  12 6 y  24

2y  4 4y  8
a.
3x3 6 x5

5 y 2 5 y3
c.
x3
x2  9

x 2  x  12 x 2  7 x  12
c. x 
3
1

x4 x6
14
2x
x
2
2
 4

 2
d.
x 1
x 1
x  2 x  1 x  3x  2
b.
Solve each problem using proportions.
2 x
9
12

a.
b. 
5 3x  2
3 6
c.
3x  2 4 x

4
5
b.
d.
d.
4
2

y 3 y 3
e. As a consulting fee, Mr. Visconti charges $90.00
per day. Find how much he charges for 3 hours of
consulting. Assume an 8-hour work day.
f. Find the unknown side for the following similar
triangles.
10 in
x
2x
 2
x2 x 4
8 in
x
18
 2
c. x  3 x  9
d.
Solve each rational equation. Check for extraneous
solutions.
x 1 x  2
5
4

2

a.
b.
3
7
x2 x2
Add the rational expressions. Write your answer in simplest
form.
15 5

4x 4x
5x 1
3x

x  7 x  12 x  3
2
Unit 8
6 in
a.
b.
2
Multiply the rational expressions. Write your answer in
simplest form.
a.
Subtract the rational expressions. Write your answer in simplest
form.
12
3

x  5 x  24 x  3
2
3 in
5 in
x in
Simplify each complex fraction.
2 1

a. x x
2
x
1
b.
x
y
Solve the following Absolute Value Inequalities. Graph
the answer and write in interval notation.
a. 2 x  3  5
x2
1
y2
b. 6 x  3
1 3

x 2x
c. 1
3

3x 4 x
2 3

x y
d.
1
5
xy
Unit 9
Write each Compound Inequality in inequality form and
interval notation and then graph.
a. x  5  2 and 2x  3  5
c. x  2  4
Graph the following Linear Inequalities.
a. y  3 x  5
b. 2 x  3 y  6
b.  x  7  5 or 3x 1  13
Graph the following Systems of Linear Inequalities.
c. 1 
x3
2
4
 y  3x  2
y  x  4
a. 
Solve the following Absolute Value Equations.
a. 2 x  3  5  7
b.
x3
 4
2
c. x  3  2 x  4
x  4 y  8
x  5

b. 