Download Test 3 Review

Test 3 Review
Cumulative Review, Tests 1 to 3
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Main Topics
• Combining terms
• Solving equations by addition
• Solving equations by multiplication
• Equations with fractions and decimals
• Equations with no solutions
• Equations that are identities
• Word Problems
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Combining Terms
• In order to combine terms, the terms must have the same
powers of the same variables
– We can combine 3x and 4x, but not x and x2
– We can combine 3xy and 4xy, but not 3x and 4y, nor 4yx2
• In order to combine, just add the coefficients (the numbers
multiplying the variables)
– To combine 3x and 4x, we have x(3+4) = 7x
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Using Addition to Solve Equations
• We can always add (or subtract) the same thing to both sides
of an equation and keep the equation the same:
equivalent equations
• 3x + 2 = 4x
– Subtract 3x from each side
– Get 3x – 3x + 2 = 4x – 3x
– Or: 2 = x
• Can add either variables or numbers
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Using Multiplication to Solve Equations
• We can always multiply (or divide) both sides of an equation
by the same thing and keep the equation the same
– If 3x = 6 then x = 6/3 = 2
We have to be careful to never divide by 0!!!
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Equations with Fraction
• It is often easier to get rid of the fractions by multiplying by the
least common multiple (the common denominator)
• Example:
3x/2 = 1/4 x + 5
Multiply both sides by 4:
3x(2) = x + 20
Now solve for x:
6x = x + 20
5x = 20
x=4
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Equations with Decimals
• It is often best to get rid of the decimals
• Example:
0.1x = 0.05x – 0.12
This is the same as:
10x = 5x – 12
Which becomes:
5x = 12
x = 12/5
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Equations with No Solution
• Sometimes we find that we get an equation that, upon
simplification, becomes something like 3 = 2;
we say that these equations have no solution
• Example:
3(2x – 5) = 6x – 4
simplifying:
6x – 15 = 6x – 4
Adding –6x to both sides gives
-15 = - 4: this equation has no solution!
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Equations that are Identities
• Sometimes we get an equation of the form x = x; this equation
is true for any value of x
We call this an identity
• Example:
3x – 4 = (1/2) ( 6x – 4) – 2
3x – 4 = 3x – 2 – 2
3x – 4 = 3x – 4
3x = 3x
x = x; holds for all x, an identity
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Sample Word Problems
• Two consecutive integers add to 11. What are the numbers?
– Consecutive numbers are n and n + 1
– Consecutive even numbers are n and n + 2
– In this case, n + n + 1 = 11, 2n = 10, n = 5;
– The numbers are 5 and 6
• If a board that is 10 feet long is cut into two pieces and one
piece is 2 feet longer than the other, how long are the two
pieces?
One piece is x, one is x + 2
x + x + 2 = 10, x = 4; the two pieces are 4 and 6 feet long
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Concepts from Previous Units
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Fraction Arithmetic
• Adding or Subtracting Fractions:
– Need to find lowest common denominator
– Example: 1/3 + 1/5 = 5/15 + 3/15 = 8/15
• Multiplying Fractions:
– Multiply numerator and denominator
– Example: 1/5 x 3/4 = 3/20
• Dividing Fractions:
– Multiply by the reciprocal, or multiplicative inverse
– Example: 3/4 ÷ 3/5 = 3/4 x 5/3 = 15/12 = 5/4
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Simplifying or Reducing Fractions
• Use prime factorization to find common factors then eliminate
multiples of 1
• Example:
90/36 = ?
90 = 2 x 3 x 3 x 5
36 = 2 x 2 x 3 x 3
90/36 = (2 x 3 x 3 x 5)/ (2 x 2 x 3 x 3)
Remove 2/2, 3/3, 3/3, and are left with
90/36 = 5/2
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Multiplicative and Additive Inverses
• A number plus its additive inverse is zero:
5 + (-5) = 0
-5 is the additive inverse of 5
• A number times its multiplicative inverse is 1
5 x (1/5) = 1
1/5 is the multiplicative inverse of 5
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Decimal Arithmetic
• Adding or subtracting: line up the decimal point:
3.14 + 0.12:
3.14
+0.12
3.26
• Multiplying: Multiply then count the number of decimals:
0.012 x 0.4, number is 48, have 4 decimal places
0.0048
• Dividing: Multiply by powers of 10 to eliminate the decimal in
the denominator: 0.12 / 0.003 = 120/3 = 40
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Combining Positive and Negative Numbers
• Positive, move right on the number line
• Negative, move left on the number line
• 5–3=2
• 7 – 9 = -2
• The sign goes with the larger number
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Multiplying and Dividing Negative Numbers
• A minus times a minus is a plus
• A minus divided by a minus is a plus
• A minus times, or divided by a plus is a minus
• Examples:
-3 x -4 = 12
3 x -4 = -12
3 x 4 = 12
-3 x 4 = -12
• For more than 2 numbers, follow order of operations:
-2 x 3 x -4 x 6 = -6 x -4 x 6 = 24 x 6 = 144
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Terminology
• An integer is a positive or negative whole number:
-3, -4, 5, 6, etc
• A rational number is a number that can be expressed as a
fraction:
1/4, 6, 3/5, -9/4501, 0.067, √4 etc.
• An irrational number is a number that cannot be expressed
as a fraction:
√2, π, etc.
• Real numbers include all rational and irrational numbers, all
numbers you know. Numbers that are not real include the
square root of -1, for example
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Terminology, Cont
• The quotient of 5 and 4 is 5/4
• The product of 5 and 4 is 5 x 4 = 20
• Five less four is 5 – 4 = 1
• Five is three more than a number: 5 = n + 3, n = 2
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Inequalities
• Which is bigger 1/3 or 1/4?
Get a common denominator:
1/3 = 4/12, 1/4 = 3/12, 1/3 is larger than 1/4, or 1/3>1/4
• Do not get fooled by negative numbers!!
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Percent
We will be doing more of this
• Per Cent means divide by 100
3% = 3/100 = 0.03
• x% of a number is x% x number
5% of 20 = 5%(20) = (5/100)(20)= 100/100 = 1
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Properties of Arithmetic
• Commutative, for addition and multiplication:
a + b = b + a, ab = ba
Does not hold for subtraction or division
• Associative, for addition and multiplication:
a + (b + c) = (a + b) + c, (ab)c = a (bc)
• Distributive
a (b+c) = ab + bc
• Identities:
Addition: identity is 0, a + 0 = a
Multiplication: identity is 1, a x 1 = a
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Mixed Numbers
• Addition: Can add the whole numbers then the fractions, or
can convert to improper fractions
3 1/2 – 2 5/8 = (3 – 2) + (1/2 – 5/8) = 1 – 1/8 = 7/8, or
3 1/2 - 2 5/8 = 7/2 – 21/8 = 28/8 – 21/8 = -7/8
• Multiplication: Convert to improper fractions then multiply
3 1/2 x 2 5/8 = 7/2 x 21/8 = (7x 21) / (2 x 8) = 147/16
If you want, can convert back: 147/16 = 9 3/16
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Order of Operations
• PEMDAS
• Parentheses
• Exponents
• Multiplication and Division
From Left to Right
• Addition and Subtraction
• Remember: -22 ≠ (-2)2
Example:
- 4 + 3x5÷2 = - 4 + 15 ÷2 = - 4 + 7 1/2 = 3 1/2
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Evaluating Expressions
• Evaluate 3 (-x + 4)2 + 5y when x = -1 and y = -2
• Follow order of operations:
3(-(-1) + 4)2+ 5(-2) = 3(1+4)2 -10 = 3(5)2 -10 = = 75 – 10 = 65
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Verifying Solutions to Equations
• Is x = -4 a solution of 3x – 6 = 2?
3(-4) – 6 = -12 – 6 = -18 ≠ 2, NO
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Absolute Value
• Always positive
• The distance from zero
| -4| = 4
| ½| = ½
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