Download Unit 2 MM Algebra

Chapter 2
Algebra
Objectives
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Solve linear equations
Solve mixture problems
Solve rational equations
Perform formulae manipulation
Evaluate problems using ratios and
percents
 Solve percent problems
© 2010 Delmar, Cengage Learning.
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Objectives (cont’d.)
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Use the properties of exponents
Use scientific notation
Evaluate significant digits
Use the scientific calculator to evaluate
expressions
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Solving Linear Equations
 If the product of two numbers is 1, they are
reciprocals
• The reciprocal of 1 ⁄ 7 is 7
–.
 Like terms have the same variable and the
same exponent
• Can be combined: 5x + 3x = 8x
© 2010 Delmar, Cengage Learning.
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Solving Linear Equations (cont’d.)
 Whatever operation is performed on one
side must also be done to the other side
 When solving any equation, the goal is to
isolate the variable
• Solve: 2x − 6 = 20
– Add 6 to both sides
– Divide both sides by 2
– Simplify: x = 13
© 2010 Delmar, Cengage Learning.
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Solving Linear Equations (cont’d.)
 Distributive property:
• a(b + c) = ab + ac
 Commutative property:
• a+b=b+a
• a×b=b×a
 Associative property:
• (a + b) + c = a + (b + c)
• (a × b) × c = a × (b × c)
© 2010 Delmar, Cengage Learning.
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Mixture Problems
 A 3% solution is needed
• Only 30 fl oz of a 4% solution is in stock
• How much “neutral” solution should be added
to 30 fl oz of the 4% solution?
x + 30
0%(x) + 4%(30) = 3%(x + 30)
0.00(x) + 0.04(30) = 0.03(x + 30)
1.2 = 0.03x + 0.9
0.03x = 0.3
x = 10
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Rational Equations
 Equation containing rational expressions
• Example:
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Formulae Manipulation
 Sometimes we need work with formulas
that do not have many numbers
• Solve for A:
.
© 2010 Delmar, Cengage Learning.
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Ratios and Proportions
 Ratios can be written three ways:
• 1 to 2
• ½
• 1:2
 Ratios are in proportion if they are
equivalent to each other:
• 2/3 is proportional to 8/12
–.
© 2010 Delmar, Cengage Learning.
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How to Calculate: Ratios and
Proportions
 Cross multiplication:
 When solving proportions:
• Components on left-hand side must be set up
in the same order as components on righthand side:
–.
© 2010 Delmar, Cengage Learning.
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Solving Percent Problems (cont’d.)
 Percents should be written as decimals
• 35% of what number is 21?
.35 × x = 21
.35x = 21
21 ÷ 0.35
x = 60
 Proportional formula:
• When using this method do not use decimals
© 2010 Delmar, Cengage Learning.
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Properties of Exponents
 Product rule: exponentials are used to
represent repeated multiplication
• .
 Quotient rule:
• .
• .
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Properties of Exponents (cont’d.)
 Power rule for fractions:
• .
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Properties of Exponents (cont’d.)
 Negative exponent rule:
• .
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Properties of Exponents (cont’d.)
 Negative exponent rule for fractions:
• .
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Properties of Exponents (cont’d.)
 There is no exponent rule for adding
exponentials
• .
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Scientific Notation
 Used when dealing with very large or very
small numbers
• .
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How to Calculate: Significant Digits
 Significant digits tell about the accuracy of
a measurement
• Rule 1: Determining whether a digit is
significant:
– All nonzero digits are significant
– Zeros are significant if they are on the right side of
a decimal number
– Zeros are significant if they are between two
significant digits
© 2010 Delmar, Cengage Learning.
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How to Calculate: Significant Digits
(cont’d.)
 Rule 2: Determining whether a zero is not
significant:
• A zero is not significant if it is on the right side
of a whole number
• A zero is not significant if it is on the left side
of a number
© 2010 Delmar, Cengage Learning.
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Using the Scientific Calculator
 When using the scientific calculator, keep
order of operations in mind
• PEMDAS
• Key is used to enter expressions that contain
exponents:
• To enter a negative number, enter the number
first and then enter the +⁄− key
© 2010 Delmar, Cengage Learning.
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Using the Scientific Calculator (cont’d.)
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• .
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Using the Scientific Calculator (cont’d.)
.
• .
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• .
.
• .
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Summary
 If the product of two numbers is 1, the
numbers are reciprocals
 When solving an equation, the goal is to
isolate or get the variable (x) by itself
 When setting up proportions, components
on both sides of equal sign must be set up
the same
 Percent problems can be solved by setting
up an equation or by using a proportion
© 2010 Delmar, Cengage Learning.
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Summary (cont’d.)
 The six rules for exponents are:
•
•
•
•
•
Product rule
Quotient rule
Power rule
Negative exponent rule
Exponent rules for fractions
© 2010 Delmar, Cengage Learning.
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