Transcript Chapter 1 Powerpoint

Chapter 1
Tools of Algebra
In Chapter 1, You Will…
Review and extend your knowledge of
algebraic expressions and your skills in
solving equations and inequalities.
Solve absolute value equations and
inequalities by changing them to
compound equations and inequalities.
1-1 Properties of Real Numbers
What You’ll Learn …
To graph and order real numbers
To identify and use properties of real
numbers
Number Classification
• Natural numbers are the counting numbers.
1,2,3,4,…..
• Whole numbers are natural numbers and zero.
0,1,2,3,4,….
• Integers are whole numbers and their opposites.
….-3,-2,-1,0,1,2,3,….
• Rational numbers can be written as a fraction.
7/5, -3/2, 0, 0.3, -1.2, 9
• Irrational numbers cannot be written as a fraction.
2 , 7, 
• All of these numbers are real numbers.
Real Numbers
Rational Numbers
Integers
0.31
-3
Whole Numbers
0
-2
3
6
5
8
Irrational Numbers
10
-10
5
2
3
4/2
-
123
25
-4
0.37
0.1011001000…
Example 1: Real World Connection
Many mathematical
relationships
involving variables
are related to
amusement parks.
Which set of
numbers best
describes the values
for each variable?
a. The cost C of
admission for n people
b. The maximum speed s
in meters per second
on a roller coaster of
height h in meters
c. The park’s profit (or
loss) P in dollars for
each week w of the
year
Example 2: Graphing Numbers on a
Number Line
-3
2
1.7
5
Change to
Decimals
Example 3: Ordering Real Numbers
An inequality is a mathematical
sentence that compares the value of
two expressions using an inequality
symbol.
<
<
Less than
Less than or equal to
Compare
> Greater than
>
- 0.25 and - 0.01
Greater than or equal to
Example 4: Finding Inverses
The opposite or
additive inverse
of any number a is
–a. The sum of
opposites is 0.
Find the opposite of
a. -3.2
b. 400
c. 1/5
The reciprocal or
multiplicative
inverse of any
nonzero number a is
1/a. The product of
reciprocals is 1.
Find reciprocal of
a. -3.2
b. 400
c. 1/5
Properties of Real Numbers
Property
Addition
Multiplication
Closure
a +b is a real number
ab is a real number
Commutative
a+b=b+a
Associative
(a+b)+c = a+(b+c) (ab)c = a(bc)
Identity
a+0=a, 0+a=a
a(1)=a, 1(a)=a
Inverse
a +(-a) = 1
a(1/a)= 1, a ≠0
Distributive
ab = ba
a(b+c) = ab+ac
Example 5: Identify Properties of Real
Numbers
Which property is illustrated?
a.
b.
c.
d.
6+(-6) = 0
(-4 * 1) - 2 = -4 – 2
(3+0) - 5 = 3 – 5
-5 + [2 +(-3)] = (-5 + 2) + (-3)
Example 6: Finding Absolute Value
The absolute
value of a real
number is its
distance from
zero on the
number line.
 Find the absolute
value of:
a. -4
b. 0
c. -1 (-2)
1-2 Algebraic Expressions
What you’ll learn …
 To evaluate algebraic
expressions
 To simplify algebraic
expressions
1.03 Operate with
algebraic expressions
(polynomial, rational,
complex fractions) to
solve problems.
Evaluating Algebraic Expressions
A variable is a symbol, usually a letter,
that represents one or more numbers.
An expression that contains one or more
variables is an algebraic expression or a
variable expression.
When you substitute numbers for the
variables in an expression and follow the
order of operations, you evaluate the
expression.
 Example 1- Evaluating an
Algebraic Expression
 Example 2- Evaluating an
Algebraic Expression with
Exponents
For x = 4 and y = -2
For c = -3 and d = 5
x+y÷x
 3x – 4y + x – y
 x + 2x ÷ y - 2y
 c2 - d2
 c(3 – d) – c2
 -d2 - 4(d – 2c)
Example 3 Real World Connection
 The expression -3y +61
models the percent of
eligible voters who voted
in presidential elections
from 1960 to 2000. In the
expression, y represents
the number of years since
1960. Find the
approximate percent of
eligible voters who voted
in 1988.
Simplifying Algebraic Expressions
In an algebraic expression such as
-4x +10, the parts that are added are
called terms. A term is a number, a
variable, or the product of a number and
one or more variables.
The numerical factor in a term is the
coefficient.
Like terms have the same variables
raised to the same powers.
Example 4
Combining Like Terms
3k - k
5z2 – 10z – 8z2 + z
-(m – n) +2(m – 3n)
y(1 + y)- 3y2 – (y + 1)
Example 5 Finding Perimeter
3c
3x
d
y
2x
2x – y
d
3x
5x – 2y
6c – 2d
2d
3c
8c +d
6c – 2d
1-3 Solving Equations
What you’ll learn …
 To solve equations
 To solve problems by
writing equations
1.03 Operate with
algebraic expressions
(polynomial, rational,
complex fractions) to
solve problems.
Solving Equations
An equations that contains a variable may
be true for some replacements of the
variable and false for others.
A number that makes the equation true is
a solution of the equation.
Properties of Equality

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


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Reflexive Property
Symmetric Property
Transitive Property
Addition Property
Subtraction Property
Multiplication Property
Division Property
Substitution Property
a=a
If a=b then b=a
If a=b and b=c then a=c
If a=b then a+c = b+c
If a=b then a-c = b-c
If a=b then ac = bc
If a=b and c≠0 then a/c = b/c
If a=b then b can be
substituted for a in any
expression to obtain an
equivalent expression.
Steps to Solving Equations
● Simplify each side of the equation, if needed, by
distributing or combining like terms.
● Move variables to one side of the equation by using
the opposite operation of addition or subtraction.
● Isolate the variable by applying the opposite
operation to each side.
• First, use the opposite operation of addition or
subtraction.
• Second, use the opposite operation of
multiplication or division.
● Check your answer.
Example 1 Solving an Equation with a
Variable on Both Sides
13y + 48 = 8y - 47
2t – 3 = 9 – 4t
Example 2 Using the Distributive Property
3x – 7(2x – 13) = 3(-2x +9)
6(t – 2) = 2(9 – 2t)
Example 3 Solving
a Formula for One
of Its Variables
Example 4
Solving an
Equation for One
of Its Variables
Solve for h.
Solve for x.
A = ½ h (b1 + b2)
ax +bx – 15 = 0
Example 5 Real World Connection
A dog kennel owner
has 100 ft of fencing
to enclose a
rectangular dog run.
She wants it to be 5
times as long as it is
wide. Find the
dimensions of the dog
run.
Example 6 Using Ratios
The lengths of the sides
of a triangle are in the
ratio 3:4:5. The perimeter
of the triangle is 18 in.
Find the lengths of the
sides.
The lengths of the
sides of a triangle are
in the ratio 12:13:15.
The perimeter of the
triangle is 120 cm.
Find the lengths of
the sides.
Example 7 Real World Connection
Radar detected an
unidentified plane 5000
mi away, approaching at
700 mi/h. Fifteen minutes
later an interceptor plane
was dispatched, traveling
at 800 mi/h. How long
did the interceptor take to
reach the approaching
plane?
1-4 Solving Inequalities
What you’ll learn …
 To solve and graph
inequalities
 To solve and write
compound inequalities
1.03 Operate with
algebraic expressions
(polynomial, rational,
complex fractions) to
solve problems.
Properties of Inequalities
Let a, b and c represent real numbers.
 Transitive Property
If a≤b and b≤c, then a≤c.
 Addition Property
If a≤b, then a+c ≤b+c.
 Subtraction Property
If a≤b, then a-c ≤b-c.
 Multiplication Property
If a≤b and c>0, then ac ≤bc.
If a≤b and c<0, then ac ≥bc.
 Division Property
If a≤b and c>0, then a/c ≤b/c.
If a≤b and c<0, then a/c ≥b/c.
Solving Inequalities
• To solve an inequality, use the same procedure as
solving an equation with one exception. When
multiplying or dividing by a negative number, reverse
the direction of the inequality sign.
• To graph the solution set, circle the boundary and
shade according to the inequality.
• Use an open circle for < or > and closed circles for ≤ or
≥.
Example 1 Solving and Graphing Inequalities
3x – 12 < 3
6 + 5(2 - x) ≤ 41
Example 2
No Solutions or All Real Numbers
as Solutions
2x – 3 > 2(x – 5)
4(x – 3)+ 7 ≥ 4x + 1
Real World Connection
The band shown at
the left agrees to play
for $200 plus 25% of
the ticket sales. Find
the ticket sales
needed for the band
to receive at least
$500
Compound Inequalities
A compound inequality is a pair of
inequalities that are joined by the words
“and” or “or”.
Example 4 Compound Inequality
Containing And
3x -1 > -28
and
2x +7 < 19
2x >x +6
and
x -7 ≤ 2
Example 4 Compound Inequality
Containing Or
4y -2 ≥ 14
X–1 <3
or
or
3y - 4 ≤ -13
x +3 > 8
Real World Connection
The plans for a gear assembly specify a
length of 13.48 cm with a tolerance of
+0.03 cm. A machinist finds that the part
is now 13.67 cm long. By how much
should the machinist decrease the length?
1-5 Absolute Value Equations
and Inequalities
2.08 Use equations and inequalities with
absolute value to model and solve
problems; justify results. a) Solve using
tables, graphs, and algebraic properties.
Objective 1: Absolute Value Equations
The absolute value of a number is its
distance from 0 on a number line and
distance is nonnegative.
3
3
3
3
Example 1 Solving Absolute Value
Equations
2y – 4 = 12
3x + 2 = 12
Example 2 Solving Multi-Step Absolute
Value Equations
3 4w – 1 - 5 = 10
2 3x - 1 + 5 = 33
Example 3 Checking for
Extraneous Solutions
2x + 5 = 3x + 4
An extraneous
solution is an
answer that is NOT
a solution
x
=x-1
Example 4 Solving Inequalities of the
form A ≥ b
3x + 6 ≥ 12
2x – 3 > 7
Properties
Let k represent a positive real number.
 x ≥ k is equivalent to x ≤ -k or x ≥ k.
 x ≤ k is equivalent to -k ≤ x ≤ k.
Example 5 Solving Inequalities of the
form A < b
3 2x + 6 - 9 < 15
5z + 3 - 7 < 34
Real World Connection
The specifications for the circumference C
in inches of a basketball for men is 29.5 ≤
C ≤ 30. Write the specification as an
absolute value inequality.
In Chapter 1, You Should Have
Reviewed and extended your knowledge
of algebraic expressions and your skills in
solving equations and inequalities.
Solved absolute value equations and
inequalities by changing them to
compound equations and inequalities.