Download Document

-6a – 13a = -19a
-8n + 14n =
6n
10r - 19r =
-9r
5xy + 3xy = 8xy
2
9s
+
2
11s
=
2
2s
10ac + 19ac = 9ac
2n x 8 =
16n
6a x 4a =
2
24a
5e x 9e =
2
45e
28b ÷ 7 =
3
5
÷ 5=
24 ÷ 4d =
4b
2
5
6/d
x - 3.5 = 8.9 – 3x
x+ 3x -3.5 = 8.9
4x = 8.9 + 3.5
4x = 12.4
x = 3.1
2x + 20 = 12
2x =
-8
X = -4
2/5 = 4y + 16 2/5
-16 = 4y
-4 = y
24 + 4c = -2c
4c = -2c - 24
6c = -24
c = -4
12123 = ____
50 10
22 = ______
10110 2
4
6
2
2
4 =4
8
6
=
1
1
= 4=
4 256
8 - 10
36
6 =
10
4
=
2
6
Rename 2.025 as a mixed number
Let x = .025
10 (x) = (.025) 10
10x = 0.25
100 (10x) = 100 (0.25)
1000x = 25.25
Rename 2.025 as a mixed number
1000x = 25.25
10x = .25
990x = 25
x = 25/900 or 1/ 36
Rename 2.025 as a mixed number
x = 25/900 or 1/ 36
2.025 = 2 + .025
.025 = 1/36
2.025 = 2 1/36
Two-Step
Inequalities
OBJECTIVE:
Solve, graph, and check
inequalities that call for two steps
to simplify
Solve. Graph and check the solution.
2x + 20 < 12
2x < 12 -20
2x < -8
x < -4
Graph the solution.
-6
-5
-4
-3
-2
-1
Check. Substitute -4 for x.
2(-4) + 20 < 12
-8 + 20 < 12
12 < 12;
False
Therefore, -4 is not a solution.
0
-6
-5
-4
-3
-2
-1
0
Check another value. Substitute -6 for x.
2(-6) + 20 < 12
-12 + 20 < 12
8 < 12;
True
Therefore, -6 is a solution.
Try -10.
2(-10) + 20 < 12
-20 + 20 < 12
0 < 12;
True
Therefore, -10 is also a solution.
Solve. Graph and check the solution.
3a < 16 + 11a
3a – 11a < 16
-8a < 16
-8
-8
a > -2
Graph the solution.
-6
-5
-4
-3
-2
-1
Check. Substitute -2 for a.
3(-2) < 16 + 11(-2)
-6 < 16 -22
-6 < -6;
False
0
Graph the solution.
-6
-5
-4
-3
-2
-1
0
Check. Substitute -2 for a.
3(-2) < 16 + 11(-2)
-6 < 16 -22
-6 < -6; False
Therefore -2 is not a solution.
Substituting 0 for a.
3(0) < 16 + 11(0)
0 < 16 +0
0 < 16 True
Therefore 0 is a solution.
Class work. PB, p119
Homework. PB, p119-120
Multistep Inequalities
with Grouping symbols
OBJECTIVE:
 solve, graph, and check the
solution of an inequality having a
grouping symbols
Solve. Graph and check the solution.
-2 4(x + 3) ≤ 16 -2
Multiply both
sides by -2.
-2
Apply the DPMoA.
4(x + 3) ≥ - 32
4x + 12 ≥ - 32
- 12
- 12
4x ≥ - 44
4
4
x ≥ - 11
-11
-10
-9
Subtract 12 from
both sides.
Divide both sides
by 4
Graph the solution.
-8
-7
-6
-5
-4
Graph the solution.
-11
-10
-9
-8
-7
-6
-5
-4
Check the solution.
4(x + 3)
-2
4(-11 + 3)
-2
4(-8)
-2
-32
-2
16
≤ 16
Try -11for x.
≤ 16
Combine like terms.
≤ 16
Multiply.
≤ 16
Divide
≤ 16 True, so -11 is a solution.
Graph the solution.
-11
-10
-9
-8
-7
-6
-5
-4
Check the solution.
4(x + 3) ≤ 16
-2
4(-5 + 3) ≤ 16
-2
4(-2) ≤ 16
-2
-8 ≤ 16
-2
4 ≤ 16
Try x = -5.
Combine like terms.
Multiply.
Divide
True, so -5 is a solution.
Class work PB, p121
HW: PB, p121-122
Multistep Inequalities
:fractions and decimals
Pp 114-115, text
objective
 solve, graph, and check the
solution of an inequality having
fractions and decimals
Example 1.Solve. Graph and check the solution.
100 (0.12x + 0.36) ≥ 0.6 100 Multiply both
sides by 100.
Subtract 36 from
12x + 36 ≥ 60
- 36 - 36
both sides.
≥ 24
12
x ≥ 2
12x
12
-3
-2
-1
0
0.12x + 0.36 ≥ 0.6
Divide both sides
by 12.
Graph.
1
2
3
4
Substitute -2 for x,
then evaluate.
0.12x + 0.36 ≥ 0.6
0.12(2) + 0.36 ≥ 0.6
Substitute -2 for x,
then evaluate.
Multiply.
Add.
0.24 + 0.36 ≥ 0.6
True, so 2 is a solution.
0.60 ≥ 0.6
0.12x + 0.36 ≥ 0.6
0.12(4) + 0.36 ≥ 0.6
Try 4 for x, then
evaluate.
Multiply.
Add.
0.48 + 0.36 ≥ 0.6
0.84 ≥ 0.6 True, so 4 is also a solution.
Class work PB, p125
HW: PB, p125-126
Compound inequalities
pp 116-117, text
OBJECTIVE:
 graph and find the solution of
compound inequalities
Graph: x > 3 and x < 7.
Graph on the number line.
x>3
0
2
1
x<7
4
3
6
5
8
7
Graph on the same
number line.
The solution set of the
compound inequality in
shortened form is:
10
9
11
Solution.
{x | 3 < x < 7}
Graph: z ≤ -2 or z ≥ 4.
Graph on the number line.
z ≤ -2
-3
-1
-2
z≥4
1
0
3
2
5
4
7
6
8
Graph on the same
number line.
The solution set of the
compound inequality in {z | z ≤ -2 or z ≥ 4}
shortened form is:
Class work. PB, p127
Homework. PB, p127-128
Polynomials
pp 124-125, text
OBJECTIVE:
 define a polynomial
 classify a polynomial by the number
of its terms
 simplify polynomials
Do You Remember?
variable
Algebraic
Expressions
A symbol, usually a letter,
used to represent a
number
Expressions that contain
variables, numbers, and
operation symbols
constant A term that doesn’t have
variables
Do You Remember?
exponent It tells how many times a
number or variable called
the base is used as a factor.
term
A __ of an algebraic
expression is a number, a
variable, or the product of a
number and one or more
vaeiables.
Remember…
 A monomial is an expression that is
a number, a variable, or the product
of a number and one or more
variables with nonnegative
exponents.
 examples:19, m, 7a2, 13xy, 1/4
abc10
Monomials that are real numbers
are called constants.
Remember…
 A polynomial is a monomial or the
sums and/or differences of two or
more monomials.
 Each monomial in a polynomial is
called a term.
 Polynomials can be classified by
their number of terms when they are
in simplest form.
Remember…
Types of Polynomial
Name
Number of Terms
Monomial 1(mono means one)
Binomial
2(bi means two)
Trinomial
3(tri means three)
Examples
2n, 4x3, r, 7, 6x2y5
2x + 8; 3b + p; 2a2 – 8b2
4n + b + c; 2x2 + 8x - 3
Remember…
Classifying Polynomials; before
classifying a a polynomial make sure it
is in its simplest form.
Example.
Simplify: x2 + 2x + 1 + 3x2 – 4x.
Then classify it.
Remember…
Example. What kind of a polynomial is
x2 + 2x + 1 + 3x2 – 4x?
x2 + 2x + 1 + 3x2 – 4x
4x2 – 2x + 1
Combine like terms.
Classify.
4x2 – 2x + 1 is a trinomial because it has 3
terms.
Class work. text, p125
Homework. PB, p139-140
Modeling Polynomials
pp 128-129, text
OBJECTIVE:
 use Algebra tiles to model polynomials
Algebra Tiles
= x2
=x
= -x2
= -x
=1
= -1
Examples of polynomials and their models.
x2 - 4
-3x2 + 2x +1
Write the polynomials modeled by each
set of Algebra tiles.
4x2 + 7x
3x2 + 3x - 5
-2x2 – 2x + 9
-3x2 + 2x - 1
If a polynomial is not in simple form, model it
with Algebra tiles then combine like tiles.
Example. Simplify 3x2 – 2x – 4 + x2 + 3x.
Model the polynomial.
Then rearrange the tiles so
the like ones are next to
each other.
Create zero pairs, (an x
tile and a –x tile, and
other opposites).
The simple form is
4x2 + x – 4.
Class work. text, p129
Homework. PB, p143-144
Add Polynomials
pp 130-131, text
OBJECTIVES
:
 model the addition of polynomials
 add polynomials algebraically
Algebra Tiles
= x2
=x
= -x2
= -x
=1
= -1
Example. Add 3x2 – 4x + 5 and 2x2 – x – 3.
Step 1. Model each polynomial.
3x2 - 4x + 5
+
-2x2 - x - 3
Step 1. Model each polynomial.
3x2 - 4x + 3
2x2 - x - 3
Step 2. Put the same tiles next to each other.
Step 2. Put the same tiles next to each other.
Step 3. Create zero pairs from opposite tiles.
Step 3. Create zero pairs from opposite tiles.
Step 4. Name the remaining tiles for the answer.
x2 - 5x + 2
Polynomials can be added algebraically, in either
horizontal or vertical form.
To add polynomials horizontally,use the Commutative
and Associative properties to group and combine like
terms
Example. Add 2x2 + 11x + 9 and 3x2 – 6x
(2x2 + 11x + 9)
+
(3x2 - 6x)
2x2 + 11x + 9
+
3x2 - 6x
2x2 + 3x2 + 11x – 6x + 9
5x2
+
5x
+9
Remove parentheses.
Use the APA and CPA to
group and combine like
terms
Answer.
To add polynomials vertically, arrange like terms in
columns and add the columns separately.
Example. Add 4x2 + 3xy – 9y2 and 6x2 – 7y2
4x2 + 3xy – 9y2
+
6x2
- 7y2
10x2
+ 3xy- 16y2
Arrange like terms in
columns
Answer.