DMAT 0093 Review

1
DMAT 0093Review

• By Diana Moore
• El Centro College

2
DMAT 0093, Objectives

• DMAT 0093 has 8 course objectives. These
  objective are listed on the course syllabus and
  correspond to the course description stated in
  the college catalog.
• The prerequisite for course is DMAT 0091 or an
  adequate assessment test score.

3
DMAT 0093, Objective 1

• Solve first-degree equations and absolute value
  equations and inequalities and demonstrate this
  ability by solving word problems.

4
Solve first-degree equations.
Use the both property of equality. Solve the
equation. 3(x 4) 6 5(x 3)
distribute 3x 12 6 5x 15
addition 3x 12 (5x) 5x 9
(5x) simplify 8x 12 9
addition 8x 12 (12) 9
(12) simplify 8x
4 division 8
8 simplify x 0.5
The solution is x 0.5
5
Solve first-degree inequalities.
Solve the inequality 8x 1 lt 3x 9
Add 8x 1 (3x) lt 3x 9
(3x) Simplify 5x 1 lt
9 Add 5x 1 (1) lt 9 (1) Simplify 5x
lt 10 Divide 5 5 Simplify
x lt 2
Graph the solution on a number line.
6
Solve first-degree inequalities.
Solve the inequality 5 2x gt 3
Add 5 2x (5) gt 3
(5) Simplify 2x gt 2 Divide 2
2 Simplify (change sign) x lt 1
Graph the solution on a number line.
7
Solve absolute value equations.
Solve the equation 2x 4 10
Solve both equations 2x 4 10
or 2x 4 10 2x 14
2x 6 x 7
x 3 Solution
x 7 or x 3
8
Solve absolute value inequalities.
Solve the inequality x 3 gt 2
Solve both inequalities x 3 lt 2
or x 3 gt 2 x lt 5
x gt 1
Solution x lt 5
or x gt 1
Graph the solution on a number line.
9
Solve absolute value inequalities.
Solve the inequality 2x 1 lt 5
Solve both inequalities 5
lt 2x 1 lt 5 add 1 6 lt
2x lt 4 divide by 2 3 lt
x lt 2 Solution 3 lt x lt 2
Graph the solution on a number line.
10
Solve word problems.
The perimeter of a rectangle is 48 in. The width
is 3 times the length. Find the dimensions.
x length 3x width
2 Length 2 width perim. 2(x) 2(3x)
48 2x 6x 48
8x 48
x 6
x 6 3x 18
The length is 6 in. The width is 18 in.
11
DMAT 0093, Objective 2

• Apply the rules for exponents in performing
  operations with polynomials, factor polynomials,
  and use the zero-product to solve equations.

12
Apply the rules of exponents.
Basic Laws of exponents xmxn xmn
xm xm-n xn (xm)n xmn (xy)m xmym
Examples x2x8 x10 x6 x4 x2
(x3)5 x15 (xy)3 x3y3
13
Apply the rules of exponents.
Simplify the expression (5x4)(6x9)
Multiply the coefficients5(6) 30 add the
exponents4 9 13 30x13
14
Apply the rules of exponents.
Simplify the expression 8x7y6
2x3y2
divide the coefficients8 / 2 4 subtract the
exponents 7 - 3 4 6 2 4
4x4y4
15
Apply the rules of exponents.
Simplify the expression (5x2y4)3
distribute the exponent (5)3(x2)3(y4)3
125x6y12
16
Apply the rules of exponents.
Simplify the expression (4x5y)2(3xy2)
distribute the exponent (4)2(x5)2(y)2
(16x10y2)(3xy2) multiply the coefficients16(3)
48 add the exponents10 1 11 2 2 4
48x11y4
17
Perform the operations with polynomials.
Find the sum of polynomials 3x2 5x 7 and
x2 x 4.
(3x2 5x 7) (x2 x 4) Add like
terms The sum is 4x2 4x 3
18
Perform the operations with polynomials.
Find the difference between the polynomials 3x2
5x 7 and x2 x 4.
o Change to addition
(3x2 5x 7) ( x2 x 4)
Add like terms The difference is 2x2 6x 11
19
Perform the operations with polynomials.
Find the product of the polynomials 3x2 5x 7
and x2 x 4.
3x2
5x 7 x2 x
4 x2(3x2 5x 7) 3x45x3 7x2 x(3x2 5x
7) 3x3 5x2 7x 4(3x2 5x 7)
12x220x-28
Add like terms 3x4 2x3 27x - 28
20
Perform the operations with polynomials.
Find the quotient of the polynomials x2 3x 2
and x 5.
2
x
x 5 x2 3x 2
x2 x x
- -
x(x 5) x2 5x
-2x -2 x
Subtract -2x - 2

-2(x 5) -2x - 10
Subtract 8
quotient x 2 remainder 8
21
Perform the operations with polynomials.
Use a special product formula to multiply the of
binomials 5x 3y and 2x 4y.
(5x 3y)(2x 4y) F O
I L (5x)(2x) (5x)(4y)
(3y)(2x) (-3y)(4y) 10x2 20xy
6xy 12y2 10x2 14xy 12y2
22
Perform the operations with polynomials.
Use a special product formula to multiply the of
binomials 3x 5y and 3x 5y.
(3x 5y)(3x 5y) A2 B2
(3x)2 (5y)2 9x2 25y2
23
Perform the operations with polynomials.
Use a special product formula to multiply the of
binomials (3x 5)2.
(3x 5)2 A2 2AB
B2 (3x)2 2(3x)(5) (5)2 9x2
30x 25
24
Perform the operations with polynomials.
Use a special product formula to multiply the of
binomials (x 7y)2.
(x 7y)2 A2 2AB
B2 (x)2 2(x)(7y) (7y)2 x2
14xy 49y2
25
Factor polynomials.
Factor the polynomial 16x5y3 24x3y2 4x2y
What is the GCF? ________
4x2y
Factor the polynomial.
4x2y(4x3y2 6xy 1)
26
Factor polynomials.
Factor the polynomial 5x(2x 3y) y(2x 3y)
What is the GCF? ________
2x 3y
Factor the polynomial.
(2x 3y)(5x y)
27
Factor polynomials.
Factor the polynomial 4x2 4xy 3y2
Factor the polynomial. 4x2 2xy 6xy 3y2
middle term (4x2 2xy) (6xy 3y2) group
terms 2x(2x y) 3y(2x y) factor
groups (2x y)(2x 3y) factor out
GCF
Which factoring method should be used?
_________________
factor by grouping
28
Factor polynomials.
Factor the polynomial x2 12x 20
Which factoring method should be used?
_________________
Factor the polynomial. ( )(
) parenthesis
trial error
x x 1st terms are x
2 10 2nd terms are
factors of 20
29
Factor polynomials.
Factor the polynomial 9x2 24xy 16y2
Which factoring method should be used?
_________________
Factor the polynomial. (
)2 parenthesis
special product A2 2AB B2
(A B)2
3x
A 3x
4y
B 4y
30
Factor polynomials.
Factor the polynomial 9x2 25y2
Which factoring method should be used?
_________________
Factor the polynomial. ( )(
) parenthesis
special product A2 B2 (A
B)(A B)
3x 3x
A 3x
5y 5y
B 5y
31
Factor polynomials.
Factor the polynomial 8x3 27
Factor the polynomial. ( )
Which factoring method should be used?
_________________
special product A3 B3 (A -
B)(A2 AB B2)
2x (2x)2 (2x)
A 2x
- 3 (3) (3)2
B 3
(2x 3)(4x2 6x 9)
32
Factor polynomials.
Factor the polynomial x3 64y3
Factor the polynomial. ( )
Which factoring method should be used?
_________________
special product A3 B3 (A
B)(A2 AB B2)
4y (4y) (4y)2
B 4y
x (x)2 - (x)
A x
(x 4y)(x2 4xy 16y2)
33
Use zero-property to solve equations.
Solve the equation x2 9x 10 0
Factor the polynomial x2 9x 10 0 (x
1)(x 10) 0
Solve the equations x 1 0 or x 10
0
x -10
x 1
or
34
DMAT 0093, Objective 3

• Perform operations with rational expressions and
  solve equations with rational expressions.

35
Perform operations with rational expressions.
Add 2 3 . x2 8x
15 x 5

( ) ( ) fill in the blanks
x 3x 3
2 3
factor the(x 3)(x 5) x 5
denominator
11
2 3x 9
multiply(x 3)(x 5) (x 3)(x 5)
add like terms
_3x 11 (x 3)(x 5)
36
Perform operations with rational expressions.
Subtract 9 _ 2 .
9x 6 3x 2
( ) ( ) fill in the blanks
11
33
9 -2
factor the3(3x 2) 3x 2
denominator
9 -6
multiply3(3x 2) 3(3x 2)
add like terms
1 / /
reduce
3 3(3x 2)
1 3x 2
37
Perform operations with rational expressions.
Multiply x 3 3x 6 .
x2 6x 8 x2 9
1 1
Cross cancel
x 3 3 (x 2) .
(x 2)(x 4) (x
3)(x 3) factor
3 (x 4)(x 3)
multiply
38
Perform operations with rational expressions.
Divide x2 1 x 1 . 4x
6 2x 3
x2 1 2x 3 change
to 4x 6 x 1
reciprocal
(x 1)(x 1) 2x 3 factor
2(2x 3) x 1
x 1 2
multiply
39
Solve equations with rational expressions.
Solve x 1 3 . x2 4
x 2

What is the LCD? (x 2)(x 2)
40
Solve equations with rational expressions.
Solve x 1 3 . x2 4
x 2

Fill in the blanks
x 1 3 . (x
2)(x 2) x 2

(x 2)(x 2)
(x 2)(x 2)
x 1 3(x 2) x 1 3x 6
2x 7 x 3.5
Solve the equation
41
DMAT 0093, Objective 4

• Perform operations and solve equations with
  expressions involving integral and rational
  exponents and radicals.

42
Perform operations with expressions involving
integral exponents
Basic Laws of exponents xmxn xmn
xm xm-n xn (xm)n xmn (xy)m xmym
Examples x2x8 x10 x6 x4 x2
(x3)5 x15 (xy)3 x3y3
43
Perform operations with expressions involving
integral exponents
Basic definitions of exponents x0 1
x-m 1 . xm
Examples 80 1 (2x)0 1 x-5
1 . X5
44
Perform operations with expressions involving
integral exponents
Simplify the expression (5x-4)(6x9)
Multiply the coefficients5(6) 30 add the
exponents-4 9 5 30x5
45
Perform operations with expressions involving
integral exponents
Simplify the expression (7x-2)(3x-5)
multiply the coefficients7(3) 21 add the
exponents-2 -5 -7 21x-7 Simplify
21 . 1 . 1 x7
21 x7
46
Perform operations with expressions involving
integral exponents
Simplify the expression 8x5y6
2x7y2
divide the coefficients8 / 2 4 subtract the
exponents 5 - 7 -2 4x-2 y4 6 2
4 Simplify 4 . y4 4y4 .
1 x2 x2
47
Perform operations with expressions involving
integral exponents
Simplify the expression x5 .
x-8
subtract the exponents5 (-8) 13 x13
Simplify x13
48
Perform operations with expressions involving
integral exponents
Simplify the expression (4x5y)-2(xy5)
distribute the exponent (4)-2(x5)-2(y)-2
(x-10y-2)(xy5) 16 add the exponents 10 1
9 x-9y3 2 5 3 16
y3 . 16x9
Simplify
49
Perform operations with expressions involving
rational exponents
Simplify the expression (6x y )2
1/2 3/2
distribute the exponent (6)2 (x1/2)2 (y3/2)2
36xy3
50
Perform operations with expressions involving
rational exponents
Simplify the expression (x y )(x y
)
1/2 1/4 1/3 2/3
add the exponent 1 1 3 2 52 3
6 6 1 2 3 8 114
3 12 12 x y
5/6 11/12
51
Perform operations with expressions involving
rational exponents
Simplify the expression x y .
x y
1/2 1/4 1/3 2/3
Subtract the exponent 1 _ 1 3 - 2 12
3 6 6 1 _ 2 3 - 8
-54 3 12 12
1/6 -5/12 x y
52
Perform operations with expressions involving
radicals.
Simplify the expression 25x2y6z10
Find the square root of 25 Divide all exponents
by 2
5xy3z5
53
Perform operations with expressions involving
radicals.
Simplify the expression 32x4y5z9
x4y4z8 y z
16 2
4x2y2z4 2yz
Factor out the perfect square.
Factor out exponents divisible by 2.
Simplify the perfect square expression.
54
Perform operations with expressions involving
radicals.
Simplify the expression 64x12y6z9
3
Find the cube root of 64 Divide all exponents by 3
4x4y2z3
55
Perform operations with expressions involving
radicals.
Simplify the expression 32x4y5z9
3
3
3
x3y3z9 x y2
8 4
3
2xyz3 4xy2
Factor out the perfect cube.
Factor out exponents divisible by 3.
Simplify the perfect cube expression.
56
Perform operations with expressions involving
radicals.
Simplify the expression 3x5y
27x2y5
4
4
81x7 y6
x4y4 x3y2
81
4
3xy x3 y2
Multiply the radical expressions.
Simplify the expression.
57
Perform operations with expressions involving
radicals.
Simplify the expression (2 3 )(6 - 3 )
2(4) 2(- 3 ) 6( 3 ) ( 3 )(- 3 )
8 - 2 3 6 3 - 3
5 4 3
Multiply the radical expressions.
Simplify the expression.
58
Perform operations with expressions involving
radicals.
Simplify the expression 5x
3y3
5x 3y3
15xy 9y4
3y . 3y
15xy 3y2


Rationalize the denominator.
Simplify the expression.
59
Perform operations with expressions involving
radicals.
Simplify the expression 4
3 2
4 . 3 2
4(3 2 ) 9 4
3 2 3 2
12 4 2 9 2


12 4 2 7

Rationalize the denominator.
Simplify the expression.
60
Solve equations with expressions involving
integral exponents.
Solve the equation (3x 2)-1 5
-1 -1
(3x 2)-1 5
61
Solve equations with expressions involving
rational exponents.
Solve the equation (3x 2) 5
1/3
1/3
3 3
(3x 2) 5
3x 2 125
3x 123
x 41
62
Solve equations with expressions involving
radicals.
Solve the equation 3x 2 5
3
3 3
3
3x 2 5
3x 2 125
3x 123
x 41
63
DMAT 0093, Objective 5

• Solve quadratic equations and inequalities, real
  and complex solutions, using various methods and
  demonstrate this ability by solving word problems.

64
Solve quadratic equations involving real and
complex solutions.
Quadratic Equation ax2 bx c 0
65
Solve quadratic equations involving real and
complex solutions.
Solve the quadratic equation 5x2 3x 4 0
Use the quadratic Formula x
-(3)
2
b b2 4ac 2a
- 4(5)(-4)
(3)
2(5)
66
Solve quadratic equations involving real and
complex solutions.
Solve the quadratic equation x2 5x 7 0
Use the quadratic Formula x
-(5)
2
b b2 4ac 2a
- 4(1)(7)
(5)
2(1)
67
Solve quadratic inequalities.
Solve the quadratic equation x2
7x 12 gt 0
Find the critical numbers x2 7x 12
0 (x 3)(x 4) 0 x 3 0 , x 4
0 x 3, x 4 Write the
solution x lt 4 or x gt 3 Graph the solution
on a number line.
68
Solve quadratic inequalities.
Solve the quadratic equation
x2 16 lt 0
Find the critical numbers x2 16
0 (x 4)(x 4) 0 x 4 0 , x
4 0 x 4, x 4 Write the solution
4 lt x lt 4 Graph the solution on a number
line.
69
Solve word problems.
The area of a rectangle is 45 in2. The length is
4 more than the width. Find the dimensions.
x width x 4 length
Length times width area (x 4) . x
45 x2 4x 45 x2
4x 45 0 (x 5)(x 9) 0 x
5 or x 9
x 5 x 4 9
The width is 5 in. The length is 9 in.
invalid
70
DMAT 0093, Objective 6

• Graph lines and inequalities on the rectangular
  coordinate system and write equations of lines.

71
Graph lines on the rectangular coordinate system.
Graph the equation y -2x 3 using a table
values.
x y -2x 3 (x,y)
y -2(2) 3 y -4 3 y -1
2
A(2,-1)
y -2(0) 3 y 3
0
B(0,3)
y -2(-1) 3 y 5
-1
C(-1,5)
72
Graph lines on the rectangular coordinate system.
Graph the equation 2x y 4 using the x- and
y-intercepts.
Let y 0 2x 0 4 2x 4 x
2 x-intercept (2,0)
Let x 0 2(0) y 4 -y 4
y -4 y-intercept (0,-4)
73
Graph lines on the rectangular coordinate system.
Graph the equation y 2x - 5 using the slope
and y-intercept.
y 2x - 5 y-intercept (0,-5)
Slope m 2 move up 2 units
move right 1 unit
74
Graph inequalities on the rectangular coordinate
system.
Graph the inequality y lt x 3
Step 1 graph the line y x 3
Step 2 test point Check (0,0) 0 lt 0
3 True
Step 3 shade one side of the line
75
Graph inequalities on the rectangular coordinate
system.
Graph the inequality y gt 3x
Step 1 graph the line y 3x
Step 2 test point Check (4,4) 4 gt 3(4)
False
Step 3 shade one side of the line
76
Write equations of lines.
Given the ordered pair (4,3) and the slope m
5, find the equation.
Use the point-slope formula
y y1 m(x x1)
The equation is y 5x 17
y ( ) ( )x ( )
3 5 4
y 3 5(x 4) y 3
5x 20 y 3 (3) 5x 20 (3)
77
Write equations of lines.
Find the equation of a line parallel to y 3x
1 and passes through (4,2).
(slope intercept form)
Use the point-slope formula
The slope-intercept form of the equation is
y 3x 10
y y1 m(x x1)
y ( ) ( )x ( )
2 3 4
y 2 3(x 4) y 2
3x 12 y 2 (2) 3x 12 (2)
78
Write equations of lines.
Find the equation of a line perpendicular to y
3x 1 and passes through (4,2).
(general form)
Use the point-slope formula
y y1 m(x x1)
The general form of the equation is x 3y
10
13
y ( ) ( )x ( )
2 4
y 2 (x 4)
13
3(y 2) 3( )(x 4)
13
3y 6 x 4
79
DMAT 0093, Objective 7

• Solve systems of equations and demonstrate this
  ability by solving word problems.

80
Solve systems of equations.
Solve the system of equations. 2x 3y
17 y x 4
Which method should be used to solve this
system? substitution method
2x 3(x 4) 17 2x 3x 12 17 x
12 17 x 5
x 5
Back substitute y x 4 y (5) 4
y 9
The solution is (5, 9)
81
Solve systems of equations.
Solve the system of equations. 4x 3y
10 x 2y 3
2 3
Which method should be used to solve this
system? Addition method
4x 3y 10 x 2y 3
8x 6y 20 3x 6y 9 11x
11 x 1
Back substitute (1) 2y 3 2y 4
y 2
The solution is (1, 2)
82
Solve word problems.
Solution A is 50 acid and solution B is 30
acid. How much of each is need to make 20
gallons of 45 acid?
You need 15 gallonsof 50 acid and 5 gallons of
30 acid.
A B 20 50A 30B 45(20)
-30 1
-30A - 30B -600 50A 30B 900
A B 20 50A 30B 900
20A 300 A 15
Back Substitute B 5
83
DMAT 0093, Objective 8

• Use properties of functions and relations,
  identify relations which are functions, and give
  the domain and range of functions.

84
Use properties of functions and relations.
A relation is a set of ordered pairs.
Example (2,3), (6,1), (4,9), (3, 2)
Domain is a set of x-values.
Domain 3, 2, 4, 6
Range is a set of y-values.
Range 9, 2, 1, 3
85
Identify relations which are functions.
A function is a set of ordered pairs such that
every x-value in the domain corresponds to
exactly one y-value in the range.
Example (2,3), (6,1), (4,9), (3, 2)
Domain 3, 2, 4, 6
Range 9, 2, 1, 3
86
Identify relations which are functions.
Is the equation 4x 2y 6 a function?
YES, solve for y 4x 2y 6 2y
4x 6 y 2x 3 The function is a set
of ordered pairs (x,y) y 2x 3
87
Identify relations which are functions.
Is the equation x2 y2 9 a function?
No, solve for y x2 y2 9 y2
9 x2 y 9 x2
This equation is not a function.
88
Give the domain and range of functions.
Find the domain of the function y x2 7x 1.
The domain of all polynomial functions is the set
of all real numbers.
89
Give the domain and range of functions.
Find the domain of the function y 3x 6.
The domain of all square root functions is
solution of 3x 6 gt 0. 3x 6 gt 0 3x
gt 6 x gt 2 Domain x x gt 2
90
Give the domain and range of functions.
Find the range of the function y 3x 6.
The range of this square root functions is y gt
0. Range y y gt 0
91
Give the domain and range of functions.
Find the domain of the function 6 .
x 3
y
The domain of this rational functions is solution
of x 3 0. Domain x x 3
92
Give the domain and range of functions.
Find the range of the function 6 . x
3
y
The range of this rational functions is solution
of y 0. range y y 0
93
End of review

• Thank you for your attention.
• We hope this review has been informative.
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94
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