Download Solving Fractional Equations For each of the following, rewrite each

Solving Fractional Equations
For each of the following, rewrite each equation into a system of equations excluding the
value(s) of x that lead to a denominator of zero; then, solve the equation for x .
All of these are _________________, so cross multiply to solve.
1)
2
3

x x4
2)
x
4
x 1
3)
x3
0
x2
4)
5
11

x4 x4
 Place all non-fractional terms over 1 (this creates an equation with three or more fractions)
REMEMBER DO NOT CROSS MULTIPLY OVER AN ADDITION OR SUBTRACTION
SYMBOL – YOU CAN ONLY CROSS MULTIPLY OVER AN EQUAL SIGN!!
 Determine the least common denominator
 Multiply each term by the least common denominator and cross divide (this should
eliminate all denominators)
 Distribute to eliminate all parentheses and combine like terms to solve.
Solve the following for the indicated variable.
1
1 1
x  3 2x
x 

7
5)
6)
16
4 2
2
7
7)
m 3(m  1)

 2(m  3)
5
2
Literal Equations
Solving Literal Equations
 The same “rules” apply as solving equations
 Put a box around whatever variable you are solving for to help you
1
8) The formula for the volume of a cone is V  r 2 h . The radius, r, of the cone may be
3
expressed as
(1)
(2)
3V
h
V
3h
V
h
1 V
(4)
3 h
(3) 3
9) The equation for the volume of a cylinder is V  r 2 h . The positive value of r , in terms of
h and V , is
(1) r 
V
h
(2) r  Vh
10) Given:
(4) r 
V
2
a
c
 1  Which is an expression for x in terms of c and a?
x
x
(1) x  c  a
(2) x  c  a
11) If
(3) r  2Vh
(3) x  a  c
(4) x  a  c  1
y b
 m , then x is equal to
xa
y  b  am
m
y  b  am
(2)
m
(1)
(3)
y ba
m
(4) y  b  am
12) Solve for x in terms of a, b, and c: ax  b  c , a ≠ 0
b
ca
a
(2)
cb
(1)
ca
b
cb
(4)
a
(3)
13) If the formula for the perimeter of a rectangle is P  2l  2w , then w can be expressed as
2l  P
2
P  2l
(2) w 
2
(1) w 
14) Given:
15) If
Pl
2
P  2w
(4) w 
2l
(3) w 
a
c
 1  write an expression for x in terms of c and a.
x
a
ey
 k  t , what is y in terms of e, n, k, and t ?
n
tn  k
e
tn  k
(2) y 
e
(1) y 
(3)
(4)
n (t  k )
e
n(t  k )
y
e
y
Consecutive Integer Word Problem Let Statements
Consecutive Even Integers:
Consecutive Odd Integers:
Let x = 1st consecutive even
Let x = 1st consecutive odd
integer
integer
nd
Let x  2 = 2 consecutive even Let x  2 = 2nd consecutive odd
integer
integer
rd
Let x  4 = 3 consecutive even
Let x  4 = 3rd consecutive odd
integer
integer
16) Find two consecutive even numbers such that the sum of the smaller number and twice
the greater number is 100.
Consecutive Integers:
Let x = 1st consecutive integer
Let x  1 = 2nd consecutive
integer
Let x  2 = 3rd consecutive
integer
17) The sum of the ages of the three Romano brothers is 63. If their ages can be represented
as consecutive integers, what is the age of the middle brother?
18) Three sisters have ages that are consecutive odd integers. Find the ages if the sum of
the age of the youngest and three times the age of the oldest is five less than five times
the middle sister’s age.
19) Find three consecutive odd integers such that eight more than the sum of the first two is
equal to eleven less than three times the third.
20) Find three consecutive even integers such that the sum of twice the first and three times
the third is fourteen more than four times the second.
21) If
represents an integer, then the next consecutive integer in terms of
(1)
(2)
22) If
is
(3)
(4)
represents an odd integer, then the next consecutive odd integer in terms of
(1)
(3)
(2)
(4)
is
Coin Word Problem
23) John has $3.20 in his bank made up of nickels, dimes, and quarters. There are 3 times as
many quarters as nickels and 5 more dimes than nickels. How many of each kind are
there?
24) John has four more nickels than dimes in his pocket, for a total of $1.25. Which equation
could be used to determine the number of dimes, x, in his pocket?
(1) 0.10( x  4)  0.05 x  $1.25
(2) 0.05( x  4)  0.10( x)  $1.25
(3) 0.10(4 x)  0.05( x)  $1.25
(4) 0.05(4 x)  0.10( x)  $1.25
25) Ronnie has twice as many dimes as pennies and 3 times as many nickels as pennies. In
all, he has $1.80. How many coins of each type does he have?
26) Haley deposited $4.50 in nickels, quarters and dimes in her coin bank. The number of
dimes exceeded the number of nickels by 5, and the number of quarters was 16 less than
the number of nickels. Find the number of each kind of coin.
27) Harrison paid a bill of $4.60 with nickels, dimes and quarters. The number of nickels was
3 less than the number of dimes. The number of dimes was 5 more than the number of
quarters. How many coins of each type did he use?
Set Builder and Interval Notation
Notation used for inclusive: bracket [  This means the number is included
Notation used for exclusive: parentheses (  This means the number is NOT included
(1,5)  1  x  5
[1,5]  1  x  5
29) Which interval notation represents the set
of all numbers from 2 through 7, inclusive?
(1) ( 2,7 ]
(2) (2,7)
(3) [ 2,7 )
(4) [ 2,7 ]
31) Which interval notation represents the set
of all numbers greater than or equal to 5
and less than 12?
(1) (5,12]
(2) (5,12)
(3) [5,12)
(4) [5,12]
33) In interval notation, the set of all real
numbers greater than
and less than or
equal to 14 is represented by
(1) (6,14]
(2) (6,14)
(3) [6,14)
(4) [6,14]
35) Which interval notation describes the set
?
(1) (1,10]
(2) (1,10)
(3)
(4)
[1,10)
[1,10]
30) Which interval notation represents
?
(1) (3,3]
(2) (3,3)
(3) [3,3)
(4) [3,3]
32) Which notation is equivalent to the
inequality
?
(1) (3,7]
(2) (3,7)
(3) [3,7)
(4) [3,7]
34) Which set of integers is included in
?
(1) {0, 1, 2, 3} (3) {-1, 0, 1, 2, 3, 4}
(2) {-1, 0, 1, 2} (4) {-2, -1, 0, 1, 2, 3}
36) Which set-builder notation describes
?
(1)
(2)
(3)
(4)
37) Written in set-builder notation,
is
1)
2)
3)
4)
38) Which set builder notation describes
?
1)
2)
3)
4)
39) The set
(1)
(2)
(3)
(4)
40) The set
(1)
(2)
(3)
(4)
is equivalent to
is equivalent to
Writing the Equation of a Line
 Calculate the slope using the slope formula m 
y 2  y1
x 2  x1
 Substitute the slope for m
 Substitute either point into x and y to find the value of the y-intercept (both points given
both lie on the same line therefore you can substitute either point into
x an y)
 Write the equation of the line by substituting into m and b
Practice: Write the equation of a line with the given conditions
41) Passes through the points
42) Passes through the points
(1, 6) and (3, -4)
(5, -2) and (4, -4)
43) Passes through the points
(-3, 5) and (-5, 8)
44) Passes through the points
(4, 3) and (-5, -2)
45) Passes through the points
(3, 4) and (-4, 6)
46) Passes through the points
(-2, 3) and (4, -5)
47) Passes through the points
(2, 5) and (-7, 5)
48) Passes through the points
(5, 13) and (10, 23)
49) What is an equation for the line that passes through the coordinates
(2, 0) and (0, 3)?
3
2
(1) y   x  3
(3) y   x  2
2
3
3
2
(2) y   x  3
(4) y   x  2
2
3
 Identify each unknown quantity and represent each one with a different variable in a let
statement. READ CAREFULLY. Make the let statements accurate.
 Translate the verbal sentences into two equations
 Solve as a system of equations. Usually these problems are solved algebraically but
follow directions – you might be asked to solve them graphically.
 Check the answers in the words of the problem.
50) Together Evan and Denise have 28 books. If Denise has four more than Evan, how many
books does each person have?
51) Eldora and Finn went to an office supply store together. Eldora bought 15 boxes of paper
clips and 7 packages of index cards for a total cost of $55.40. Finn bought 12 boxes of
paper clips and 10 packages of index cards for a total cost of $61.70. Find the cost of one
box of paper clips and the cost of one package of index cards.
52) Dalton has 7 bills, all tens and twenties, that total $100 in value. How many of each bill
does he have?
53) Harold had a summer lemonade stand where he sold small cups of lemonade for $1.25
and large cups for $2.50. If Harold sold a total of 155 cups of lemonade and collected a
total of $265, how many cups of each type did he sell?
54) Galina spent $3.60 for stamps to mail packages. Some were 30¢ stamps and the rest
were 20¢ stamps. The number of 20¢ stamps was 2 less than the number of 30¢ stamps.
How many stamps of each kind did Galina buy?