Download Unit 6 Linear Inequalities

March 07, 2012
Chapter 6
1.
4x = 9.6
What is the value of x?
2.
250 + 3.5n = 670
What is the value of n?
3.
5=
4.
7.6 = -2(-3 - y)
What is the value of y?
5.
9% of a number is 54
What is that number?
q
-2
-5
What is the value of q?
6. Erica is thinking of a number. If you divide the number by 3
then subtract 13.5 the result is 2.8. Let b represent Erica's
number. Write an equation to determine this number.
7. A large pizza with tomato sauce and cheese costs $7.50, plus
$1.50 for each additional topping. A customer orders a large
pizza and is charged $16.50. How many toppings did the
customer order?
March 07, 2012
6.1 Solving Equations by Using Inverse Operations
What is the inverse operation of ADDITION?
What is the inverse operation of MULTIPLICATION?
WE CAN USE INVERSE OPERATIONS TO
SOLVE MANY TYPES OF EQUATIONS.
How do we SOLVE
x + 2.4 = 6.5
x - 3.8 = 7
What 'inverse' operation would do this easily? Why?
6.1 Solving Equations by Using Inverse Operations
How about
4x = 16
x
=7
3
Why? What are the inverse operations?
How many 'steps' were involved in solving each of
the former equations?
Do you think the 'concept' would change if the
equations got bigger? What would change?
March 07, 2012
6.1 Solving Equations by Using Inverse Operations
Can you create an equation that would require
someone to use two steps to solve?
Examples:
x
+ 5 = 12
4
Is it important to do the steps in a certain
order? Should one INVERSE OPERATON be
completed before another one?
4.5 d - 3.2 = - 18.5
6.1 Solving Equations by Using Inverse Operations
These inverse operations can be shown pictorially as well:
4.5 d - 3.2 = - 18.5
March 07, 2012
6.1 Solving Equations by Using Inverse Operations
When solving algebraic equations, you must understand the principles of inverse
(opposite) operations:
The inverse of + is _______.
The inverse of - is _______.
The inverse of x is _______.
The inverse of ÷ is _______.
Steps for Solving Equations Using Inverse Operations:
Step 1: Start with a variable
Step 2: build equation (x or ÷ before + and -)
Step 3: Working backwards, solve for the variable. (+ or - before
x and ÷)
6.1 Solving Equations by Using Inverse Operations
Ex #1: A number is divided by 4 and subtracted by 10 to get -18. What is the
variable?
Algebraically:
Ex #2: 3d + 2.2 = 8.8
March 07, 2012
6.1 Solving Equations by Using Inverse Operations
ex. A rectangle has a length 3.7 cm and perimeter 13.2cm.
w
3.7 cm
i) Write an equation that can be used to determine the width of the rectangle:
ii) Solve the equation:
iii) Verify (check) the equation:
6.1 Solving Equations by Using Inverse Operations
a) Three times a number is - 3.6
b) A number divided by 4 is 1.5
March 07, 2012
6.1 Solving Equations by Using Inverse Operations
SCIENCE
When a freighter unloads its cargo, it replaces the
mass of cargo with an equal mass of sea water.
This mass of water will keep the ship stable. The
volume of sea water added is measured in liters. To
relate the volume of water to its mass, we use the
formula for density,D:
D = M ÷ V, where M = mass, and V = volume
6.1 Solving Equations by Using Inverse Operations
Here are a couple of "trickier" examples...
ex. 7% of a number is 56.7. Write and solve an equation.
ex. 8.4 = -6(a + 2.4)
Distribute first, then solve for the variable!
March 07, 2012
6.1 Solving Equations by Using Inverse Operations
DISCUSSION
How are inverse operations used to solve an equation?
How can you verify your solution of an equation?
When you build or solve an equation, why must you
apply the operations or inverse operations to both
sides of the equation?
When you solve a two - step equation using inverse
operations,how is the order in which you apply the
inverse operations related to the order in which you
would build the end equation?
6.1 Solving Equations by Using Inverse Operations
Homework / Practice:
P. 272 #6ac, 7, 8ace, 9ac, 11abcd, 12,
14, 15, 17, 18ace, 19, 20
March 07, 2012
6.2 Solving Equations by Using Balance Strategies
Try to solve the following... what does 'a' equal?
5a + 7 = 2a + 1
What strategy or method did you use?
6.2 Solving Equations by Using Balance Strategies
Multi-Step equations use the same principles as 1
and 2 step equations:
- Use inverse operations to...
- Isolate the variable
1. Use inverse operations and collect like terms so that the
variable appears only once in the equation
2. Solve (isolate variable) using inverse operations
3. You must complete the same operations on both sides of
the equation
March 07, 2012
6.2 Solving Equations by Using Balance Strategies
Think of a set of balanced scales. The scales will
remain balanced if we do the same thing on each side.
4x + 8 = 2(7+x)
6.2 Solving Equations by Using Balance Strategies
Examples:
8n + 10 = -4(1 - 4n)
3x - 5 = -2x + 20
3a - 6a = 5a + 8
March 07, 2012
6.2 Solving Equations by Using Balance Strategies
Examples:
122
r
= 3 + 2r
2a= 4a + 7
3 5
x+7=5
6
4 4
6.2 Solving Equations by Using Balance Strategies
Really tough ones...
2 - x = 5x + 1
24
24
25 + x = 7x - 5
9
9
6
2
March 07, 2012
6.2 Solving Equations by Using Balance Strategies
Ex. Steven has a choice of 2 companies to rent a car.
Company A charges $100 per week, plus $0.25 per km driven.
Company B charges $60 per week, plus $0.45 per km driven.
Determine the distance that Steven must drive for the 2 rental
costs to be the same.
a) Model the problem with an equation:
b) Solve the problem:
c) Verify the solution:
6.2 Solving Equations by Using Balance Strategies
Assignment page 280 - 281
# 4, 7i, 9, 11ace, 12, 13, 15, 17cd, 18
March 07, 2012
6.3 Introduction to LINEAR INEQUALITIES
DISCUSS:
WHAT DO YOU THINK THE DIFFERENCE IS
BETWEEN AN EQUATION AND AN
INEQUALITY?
<
<
>
>
Less than
Greater than
Less than or equal to
Greater than or equal to
* NOTE - GOOD DISCUSSION (VISUAL) ON PAGE 288
6.3 Introduction to LINEAR INEQUALITIES
Examples:
a is less than 3
a<3
b is greater than -4
b > -4
c is less than or equal to 3
4
c≤
d is greater than or equal to -5.4
d ≥ 5.4
3
4
March 07, 2012
6.3 Introduction to LINEAR INEQUALITIES
Choosing variables and writing inequalities:
a) Contest entrants must be at least 18 years of age
b) The temperature has been below - 5 C˚ for the last week.
c) You must have 7 items or less to use the express checkout line at Sobeys.
d) Scientists have discovered over 400 species of dinosaurs.
6.3 Introduction to LINEAR INEQUALITIES
Which numbers are solutions to b ≥ - 4? (How can you tell? Strategies?)
-8
- 3.5
20/-2
-4
√8
0
- 4.5
-1
March 07, 2012
6.3 Introduction to LINEAR INEQUALITIES
Representing Inequalities Graphically:
- Use a shaded circle if the value can be equal to the number
- Use an un-shaded circle if the value is greater or less than the number
a<3
a is less than 3
-10
-9
-8
-7
-6
-5 -4
-3
-2
-1
1
0
2
3
4
5
6
7
8
10
9
b > -4
b is greater than -4
-10
-9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
c is less than or equal to 3
4
-10
-9
-8
-7
-6
-5 -4
-3
-2
-1
-9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
d ≥ 5.4
d is greater than or equal to -5.4
-10
c≤
3
4
5
6
7
8
9
10
6.3 Introduction to LINEAR INEQUALITIES
Graph each inequality on a number line.
Write 4 numbers that are solutions of the inequality
a) t > - 5
b) - 2 ≥ x
c) 0.5 ≤ a
d) y < -( 25
-3 )
March 07, 2012
6.3 Introduction to LINEAR INEQUALITIES
Assignment pages 292 - 293
# 4,5,7,8ace,9aceg,11,12,14
Try #15 if time.
6.4 Solving Linear Inequalities by Using Addition and Subtraction
Solving inequalities uses the same principles as solving equations:
x+2=8
x had to equal 6 for
equation to be true.
x + 2> 8
x has to be
greater than 6 for
this equation to
be true.
March 07, 2012
6.4 Solving Linear Inequalities by Using Addition and Subtraction
Examples:
a + 6 < 12
2y + 33 > -18 + y
You can verify your solutions by choosing numbers that fit the
inequality and replacing them in the original problem.
6.4 Solving Linear Inequalities by Using Addition and Subtraction
Match the inequality with the graph:
a) c - 2 > 2
b) 1 > r + 8
10 11 12 13 14 15 16
2
3
4
5
6
7
8
c) 8 > -5 + w
-14 -13 -12 -11 -10 -9 -8
d) 7 + m < -2
-12 -11 -10 -9 -8 -7 -6
March 07, 2012
6.4 Solving Linear Inequalities by Using Addition and Subtraction
Example 3: Jake plans to board his dog while on vacation. Boarding house A charges
$90.00 plus $5.00 per day. Boarding house B charges $100.00 plus $4.00 per day.
For how many days must Jake board his dog for boarding house A to be less expensive than
boarding house B?
a) Write an inequality
b) Solve the problem and Verify
c) Graph the problem
6.4 Solving Linear Inequalities by Using Addition and Subtraction
Practice:
Pg 298-299
# 5c,6d,8ace,9ace,14
March 07, 2012
6.5 Solving Linear Inequalities by Multiplying and Dividing
When Multiplying or Dividing inequalities the rule is:
When you multiply or divide BY A NEGATIVE
NUMBER you must REVERSE THE INEQUALITY !!
ex. -2w < 20
ex. 2h - 12 > 24
-3
ex. -4w + 5 > -3
ex. -2(3 - 1.5n) < 3(2 - n)
6.5 Solving Linear Inequalities by Multiplying and Dividing
A super-slide charges $1.25 to rent a mat and
$0.75 per ride. Sidney has $10.25, how many
rides can he go on?
March 07, 2012
6.5 Solving Linear Inequalities by Multiplying and Dividing
Practice P. 305-306 # 9, 10, 12ac, 18
Unit Review P. 308 - 309
# 1iii, 3cd, 4, 5, 8, 10d, 15, 16
& Practice Unit Exam online