Exam 1 Review:
Here is a list of topics and so
Order of Operations: Simplify the following:
- 9 2[6(3-5) + 7]
- 9 2[6(-2) + 7]
- 9 2[-12 + 7]
- 9 2[-5]
- 9 + 10
- 19
+ 5(7 - 13)- -9 +5(7 13)
- -9 + 5(-6)
- -9 30
- -39
Translate English phrases into mathematical
expressions: Dont forget to label you
variable.
- Five
less than the product of four and the sum of a number and six.
Let x = a number.
4(x + 6) - 5
What is an algebraic
expression?
An algebraic expression is a collection of numbers,
variables, operation and grouping symbols.
Note that there is no equal sign.
What is an equation?
An equation is a comparison of two algebraic expressions by an equals sign.
What is a linear
equation?
A linear equation is an equation which can be simplified to Ax +B = 0 where A and B are real numbers and A does not equal 0.
What are the three
types of equations in one variable?
Conditional equations: Equations which are true for at least on value of the variable and false for at least one value of the variable.
Identity equations: Identity equations are true for all values of the variable. The will simplify down to 0 = 0.
Contradictions:
Contradictions are always false no matter what the value of the
variable. They will simplify down to so
What are equivalent algebraic expressions? Equivalent algebraic expressions are expressions which are equal no matter what value you plug in for the variable(s).
What are equivalent
equations?
Equivalent equations are equations which have the sa
- Since the coefficient of x cannot be 0, there will always be one solution.
- An equation may have just two solutions, thus making it a conditional equation, however it will not be linear.
- 3(x + 2) = 3x + 6 is an example of a contradiction. F
- It simplifies to 0 = 0 which is always true. Therefore it is an identity equation.
- The solution for an identity equation is all real numbers. T
- Identities are true for any value of the variable. Thus all real numbers is the solution.
- If an equation has more than one solution, then it is an identity. F
- It
will be a conditional equation as it is so
= 0 is an example of a linear equation. F- There is an x squared term.
is an example of a linear
equation. T - The coefficient of x is a real number.
is an example of a linear equation. F- the variable is under the radical and is thus not a linear equation.
Evaluate algebraic expressions. Evaluate the following for x = -3 and y = 2.
Know the Real Number System.
- What are the integers?
- The whole numbers?
- The natural numbers?
- The rational numbers?
- The irrational numbers?
- The
real numbers?
List the smallest set that the following numbers belong to.
- 5 The Natural Numbers
- 0 The Whole Numbers
- -12 The Integers
- .3 The Rational Numbers
- .303303303303 The Rational Numbers
The
Irrational Numbers
The
integers
Not
a real number
What is the absolute
value of a number?
It is the distance the number is from 0 on the number line.
What are the properties of real numbers?
Commutative
property of addition
Commutative
property of multiplication
Associative
property of addition
Associative
property of multiplication
The
distributive property
The
additive identity
The
multiplicative identity
The
additive inverse
The
multiplicative inverse
The following are examples of what properties?
(3*4)(5)=5(3*4) The
Commutative Property of Multiplication
5 + 0 = 5 The
Additive Identity
5 + 0 = 0 + 5 The
Commutative Property of Addition
5 + (-5) = 0 The
Additive Inverse
[(5 + 0) +3] = 5 + 3 The
Additive Identity
[(5 + 0) + 3] = 5 + (0 + 3) The Associative Property of Addition
5x + 3x = x(5 + 3) The Distributive Property
Simplify expressions: make sure that you add like terms!
- 5(2x + 3) - (6x -7)
- 10x + 15 -6x + 7
- 4x + 22
- 2 - 4(3x + 6) + 5(8x + 7)
- 2 12x -24 + 40x + 35
- 28x + 13
- 4xy + 3x (6x 5yx)
- 4xy + 3x -6x +5xy
- 9xy 3x
Solving equations:
- x + 7 = 9 Subtract 7 from each side.
- X = 2
- x -12 = -3 Add twelve to each side
- x = 9
- -8 = 4x Divide each side by 4
- -2 = x
Subtract
5 from each side
Multiply
each side by the reciprocal, 4/3
Simplify- x = -12
Multiply
both sides by the LCD, 20
Reduce- 4(2x + 3) + 5(6x 7) = 53 Distribute
- 8x + 12 + 30x -35 = 53 Combine like terms
- 38x -23 = 53 Add 23
- 38x = 76 Divide by 38
- x = 2
Change
the subtraction to adding the opposite.
Multiply
both sides by the LCD, 21
Reduce
and distribute- 3(5x 8) + 7(-4x + -6) = 21x -168 Distribute
- 15x 24 -28x 42 = 21x 168 Combine like terms
- -13x -66 = 21x 168 Add 13x to each side
- -66 = 34x 168 Add 168
- 102 = 34x Divide
- x = 3
- .04(.3x + 2) -5( .046x + .03) = -.179 Multiply through
- .012x +.08 - .23x - 0.15 = -.179 Combine like terms
- -2.18x - .07 = -.179 Add .07
- -.218x = -.109 Divide
- x = -109/-218 Reduce
- x = .5
Applications:
Consecutive
integers
Five times the largest of 3 consecutive even integers is 4
less than three times the sum of the other two integers. What are the integers?
Let x = the first integer
X + 2 = the second integer
X + 4 = the third integer
5(x + 4) =
3(x + x + 2) -4
5x + 20 = 3(2x + 2) 4
5x + 20 = 6x + 6 4
X = 18
The integers are 18, 20 and 22.
Complementary
angles
Supplementary
angles
The sum of an angle and twice its complement is its
supplement. What is the supplement of
the angle?
Let x = the angle
90 x = the complement of the
angle
180 x = the supplement of the angle
X + 2(90 x ) = 180 x
X + 180 2x = 180 x
-x + 180 = 180 x
This is an identity and will work for any angle
Interpret English sentences and
change them into algebraic equations.
The number of women in a math 0094 class was 3 more than the
number of men. The sum of twice the
number men and three times the number of women was 49. How many women were in the class?
Let x = number of men
3 + x = the number of women
2x + 3(x + 3) = 49
X = 8
The number of women was 11.