Download Unit 2- Equations, Inequalities, and Problem Solving

MAT 100
Armstrong/Pierson
Equations, Inequalities, and Problem Solving
Unit 2
Unit Planner
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2.1
Solving Equations Using Properties
Read pages 120-127
p. 128-130
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2.2
More About Solving Equations
Read pages 130-137
p. 139
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2.3
Applications of Percent
Read pages 140-145
p. 146
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2.4
Formulas
Read pages 149-157
p. 159-160
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2.5
Problem Solving
Read pages 163-168
p. 170
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2.6
More about Problem Solving*
Read pages 172-179
p. 179-180
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2.7
Solving Inequalities
Read pages 183-192
p. 195
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Unit 2 Summary/Review
Read pages 197-205
Page 206-207
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Practice Test
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Unit 2 Test
Cumulative Review Page 208-209 #1-50 (due two days after test)
NOTES:
Unit 2 – Equations, Inequalities, and Problem Solving
2.1 – Solving Equations Using Properties
Addition Property of Equality:
Subtraction Property of Equality:
Multiplication Property of Equality:
Division Property of Equality:
“Isolate” a variable:
EX:
Isolate a:
-3 = a – 9
Name the solution:
Solve for m:
Solve for b:
What Property was used?
x + 1/3 = 3/4
What is the solution set?
36.25 + x = 48.36
What Property was used?
/6 b = 10
What Property did you use?
5
Is that the only process?
What is the solution set?
-3.25 = -2.6y
When reading your assignment, keep in mind these questions:
1. What is the difference between an expression and an equation?
2.
Equations with the same solutions are called equivalent equations. Explain why this is true.
3. A problem may ask for an answer as a solution or solution set. How do you alter the solution to make it a
solution set?
4. Are there any vocabulary words that you did not understand? Are there any questions you need to ask?
Unit 2 – Equations, Inequalities, and Problem Solving
2.2 – More About Solving Equations
Linear Equation:
Solving Multi-Step Equations:
1. Distributive Property
2. Combine Like Terms
3. Variables Both Sides
4. Properties of Equality
Solve the equation and verify the solution:
-15x + 3 = 48
Solve the equation and verify the solution:
4(x + 1) – 5x = 3
Solve the equation and verify the solution:
4x + 9 = 3x + 1
2
Isolate the variable:
3(y + 2) – 2y = 4(y – 3) – 3y
a. What happened?
b. What is the solution?
c. What is the solution set?
d. Contradiction:
Isolate the variable:
a. What happened?
b. What is the solution?
c. What is the solution set?
d. Identity:
4(x – 3) + 3x = 12(x – 1) – 5x
Unit 2 – Equations, Inequalities, and Problem Solving
2.3 – Applications of Percent
Describe how to change decimals to percents:
Describe how to change percents to decimals:
Percent Proportions
Is = %
Of
100
Tax =
Original
%
100
Part =
%
Whole
100
Increase/Decrease = %
Original
100
Tip =
Original
Commission = %
Original
100
%
100
For each example, write a proportion and solve:
What number is 5.6% of 40?
102 is 21.3% of what number?
31 is what percent of 500?
Use the pie chart on page 141. The U.S. population is predicted to be about 479 million in the year 2075. Find
the predicted number of residents in 2075 that will be in the under 13 age group.
In 2003 the tuition and fees at 4-year public colleges averaged $4694 per year. In 2008 the tuition and fees had
increased to $6585. What was the percent of increase in the tuition and fees for 4-year public colleges from
2003 to 2008? Round to the nearest percent. (Source: collegeboard.com)
A 40% discount on a winter coat amounted to a $60 savings. Find the cost of the winter coat before the
discount.
A clothing store clerk receives a 2.5% commission on all sales. What was the customer’s total if the clerk’s
commission was $9.38?
Unit 2 – Equations, Inequalities, and Problem Solving
2.4 – Formulas
You have at most 15 minutes to find the formulas from pages 149- 157 that you will be using in this section.
Formulas from business:
Formulas from science:
Formulas from geometry:
To isolate a variable in terms of other variables, use the properties of equality:
Solve for a:
P=a+b+c
Solve for h:
V = 1/3 Bh
Solve for y:
5x + 3y = 9
Solve for r2:
V = 1/3 π r 2 h
GUIDED PRACTICE Page 158-159:
11.
17.
22.
24.
28.
Unit 2 – Equations, Inequalities, and Problem Solving
2.5 – Problem Solving
Strategy:
Analyze the problem.
Define variable(s).
Form an equation.
Solve the equation.
State the conclusion.
Label/show units.
Check the result.
1. A youth soccer league had shirts printed up for each of the players. They were charged $18 per shirt
plus a one-time setup fee of $20. If the total due was $4232, how many shirts were ordered?
2. A homeowner is planning on selling his home. He wants to get $240,975 after paying a 5.5%
commission. What selling price is needed to meet his requirements?
3. A youth soccer league had shirts printed up for each of the players. They were charged $18 per shirt
plus a one-time setup fee of $20. If the total due was $4232, how many shirts were ordered?
4. The birth years of the three children in a family are consecutive even integers. The sum of the integers
is 5982. Find the year of each child’s birth.
5. A woman has 28 meters of fencing to make a rectangular kennel. If the kennel is to be 6 meters longer
than it is wide, find its dimensions.
6. The perimeter of an isosceles triangle is 24 cm. If the base is 6 cm, find the length of each remaining
side.
Unit 2 – Equations, Inequalities, and Problem Solving
2.6 – More About Problem Solving
FORMULAS
I = Prt
d = rt
(solution)(strength) (amount)(price) =
= pure mixture
Total Value
(number)(value) =
Total Value
Using a table may help in analyzing information, working with the equation,
and creating a conclusion.
1. A professor has $15,000 to invest for one year, some at 8% and the rest at 7%. If she will earn $1,110
from these investments, how much did she invest at each rate?
I = Prt
P
r
t
I
Rate 1
Rate 2
Total: $1,110
Analyze the problem.
Define variable(s).
(ABOVE)
Form an equation.
Solve the equation.
State the conclusion.
Label/show units.
2. A car leaves Peru traveling toward Albany at the rate of 55 mph. At the same time, another care leave
Alban traveling toward Peru at the rate of 50 mph. How long will it take them to meet if the cities are
157.5 miles apart?
d = rt
r
t
d
Car 1
Car 2
Analyze the problem.
Define variable(s).
Total: 157.5 miles
State the conclusion.
Label/show units.
Form an equation.
Solve the equation.
(ABOVE)
3. A bus carrying a group of camper leaves Normal, Illinois, traveling at a rate of 45 miles per hour. Half
an hour later, a car leaves Normal traveling at 65 miles per hour with camping gear that the campers
forgot. How long will it take the car to catch the bus?
d = rt
r
t
d
Bus
Car
Analyze the problem.
Define variable(s).
(ABOVE)
Form an equation.
Solve the equation.
State the conclusion.
Label/show units.
4. Cream is approximately 22% butterfat. How many gallons of cream must be mixed with milk test at
2% butterfat to get 20 gallons of milk containing 4% butterfat?
(amount)(strength) = pure mixture
amount
strength
pure mix
Cream
Butterfat
Mixture
Analyze the problem.
Define variable(s).
Form an equation.
Solve the equation.
State the conclusion.
Label/show units.
(ABOVE)
5. The owner of a store wants to make a mixture of two candies to use up 10 pounds of candy that sells
for $3.10 per pound. How many pounds of candy that sells for $2.95 per pound should be mixed with
the more expensive candy to obtain a mixture that cost $3.00 per pound?
(amount)(price) = Total Value
amount
price
Total
Cheaper Candy
Expensive Candy
Mixture
Analyze the problem.
Define variable(s).
(ABOVE)
Form an equation.
Solve the equation.
State the conclusion.
Label/show units.
6. Tickets to a concert cost $4 for students, $7 for adults, and $5 for senior citizens. Twice as many
students as adults attended the concert. If the total receipts were $875, how many students attended
the concert?
(number)(price) = Total Value
amount
price
Total
Students
Adults
Seniors
Analyze the problem.
Define variable(s).
(ABOVE)
Form an equation.
Solve the equation.
State the conclusion.
Label/show units.
Unit 2 – Equations, Inequalities, and Problem Solving
2.7 – Inequalities
Symbols:
Less than
(Strict Inequality)
is less than or equal to
Greater than
(Strict Inequality)
is greater than or equal to
Is not equal to
There are three ways to describe the solution set of an inequality:
If the solution is x ≥ 3
1. Set builder notation
2. Number line graph
3. Interval notation
EX:
Is -3 a solution of 4x + 5 ≤ -6?
Solve x – 5 ≤ 3. Write the answer in interval notation.
If the solution is x > 3
Solve each inequality, write the answer in interval notation, and graph it.
-
5
/3 x > 10
-6x ≤ 12
Graph – 3 < x ≤ 4 on a number line.
Solve -6 < 3(x + 1) ≤ 9. Write the solution set in interval notation and graph it.
A student has scores of 68%, 67%, and 72% on three exams. What percent score does he need on the last test
to earn a grade of no less than 70%?