Warm Up
Andrew is practicing for a tennis tournament and needs more tennis
balls. He bought 10 cans of tennis balls online and received a 25%
discount. The shipping cost was $5.99. Let x represent the cost of each
can.
1. Write an algebraic expression to represent the cost of the tennis
balls.
2.
Write an algebraic expression to represent the cost of the tennis
balls with the discount.
3.
Write an algebraic expression to represent the total cost of the
tennis balls with the shipping cost and the discount. Simplify the
expression.
CREATING EQUATIONS
AND INEQUALITIES
Vocabulary
• Equation: a mathematical sentence that uses an equal sign (=)
to show that two quantities are equal
• Linear equation: an equation that can be written in the form
ax + b = c
• Rate: a ratio that compares different kinds of units
Creating Equations from Context
1. Read the problem.
2. Reread the scenario and make a list of the known
3.
4.
5.
6.
quantities.
Read the statement again and look for the unknown
or the variable.
Create expressions and inequalities from the known
quantities and variable(s).
Solve the equation.
Convert the appropriate units if necessary.
Example 1: James earns $15 per hour as a teller at a bank. In
one week he pays 17% of his earnings in state and federal taxes.
His take-home pay for the week is $460.65. How many hours did
James work?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known quantities
and variable(s).
5. Solve the equation.
6. Convert the appropriate units if necessary.
Example 2: Brianna has saved $600 to buy a new TV. If the TV
she wants costs $1,800 and she saves $20 a week, how many
years will it take her to buy the TV?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known
quantities and variable(s).
5. Solve the equation.
6. Convert the appropriate units if necessary.
Example 3: Suppose two brothers who live 55 miles apart decide
to have lunch together. To prevent either brother from driving the
entire distance, they agree to leave their homes at the same time,
drive toward each other, and meet somewhere along the route.
The older brother drives cautiously at an average speed of 60
mph. The younger brother drives faster, at an average speed of 70
mph. How long will it take the brothers to meet each other?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known quantities
and variable(s).
5. Solve the equation.
6. Convert the appropriate units if necessary.
Example 4: Think about the following scenarios. In what units
should they be reported? Explain the reasoning.
1. Water filling up a swimming pool.
2. The cost of tiling a kitchen floor.
3. The effect of gravity on a falling object.
4. A snail traveling across the sidewalk.
5. Painting a room.
Example 5: Ernesto built a wooden car for a soap box derby. He
is painting the top of the car blue and the sides black. He already
has enough black paint, but need to buy blue paint. He needs to
know the approximate area of the top of the car to determine the
size of the container of blue paint he should buy. He measured the
length to be 9 feet 11 ¼ inches, and the width to be ½ inch less
than 3 feet. What is the surface area of the top of the car? What is
the most accurate area Ernesto can use to buy his paint?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known quantities
and variable(s).
5. Solve the equation.
6. Convert the appropriate units if necessary.
You Try:
To celebrate graduation, you and 4 of your closest friends have
decided to take a 5-day white-water rafting and hiking trip. During
your 5-day trip, 2 days are spent rafting. If the rafting trip covers a
distance of 60 miles and you are expected to raft 8 hours each
day, how many miles must you raft each hour?
For the hiking portion of your trip, you and your friends carry the
same amount of equipment which works out to be 35 pounds of
equipment each. For extra money, you can hire an assistant, who
will carry 50 pounds of equipment. Each assistant charges a flat
fee of $50 and an additional $22 for each mile. The total amount
you would have to pay the assistant is $512. How many miles will
your group be hiking? Is it worth hiring two assistants to help you
and your friends carry the equipment? Justify your answers.
Warm Up
Two people can balance on a seesaw even if they are different weights.
The balance will occur when the following equation
is
satisfied. In this equation, w1, is the weight of the first person, d1 is the
distance the first person is from the center of the seesaw, w 2, is the
weight of the second person, d2 is the distance the second person is
from the center of the seesaw.
1. Eric and his little sister Amber enjoy playing on the seesaw at the
playground. Amber weighs 65 pounds. Eric and Amber balance
perfectly when Amber sits about 4 feet from the center and Eric sits
about 2 ½ feet from the center. About how much does Eric weigh?
2.
Their little cousin Aleah joins them and sits right next to Amber. Can
Eric balance the seesaw with both Amber and Aleah on one side, if
Aleah weighs about the same as Amber? If so, where should he sit?
If not, why not?
INEQUALITIES
Vocabulary
• Inequalities are similar to equations in that they are
mathematical sentences, but they have infinite solutions, instead
of only having one solution.
• Symbols used:
Solving Inequalities
• Solving a linear inequality is similar to solving a linear
equation. The processes used to solve inequalities are the
same as solving linear equations
1. Read the problem.
2. Reread the scenario and make a list of the known
quantities.
3. Read the statement again and look for the unknown or
the variable.
4. Create expressions and inequalities from the known
quantities and variable(s).
5. Solve the problem.
6. Interpret the solution of the inequality in terms of the
context of the problem.
Example 1: Juan has no more than $50 to spend at the mall. He
wants to buy a pair of jeans and some juice. If the sales tax on the
jeans is 4% and the juice with tax costs $2, what is the maximum
price of jeans Juan can afford?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known quantities
and variable(s).
5. Solve the problem.
6. Interpret the solution of the inequality in terms of the context of
the problem.
Example 2: Alexis is saving to buy a laptop that costs $1,100.
So far she has saved $400. She makes $12 an hour
babysitting. What’s the least number of hours she needs to work
in order to reach her goal?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known
quantities and variable(s).
5. Solve the problem.
6. Interpret the solution of the inequality in terms of the context
of the problem.
Example 3: A radio station is giving away concert tickets. There
are 40 tickets to start. They give away 1 pair of tickets every
hour for a number of hours until they have at most 4 tickets left
for a grand prize. If the contest runs from 11:00 AM to 1:00 PM
each day, for how many days will the contest last?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known
quantities and variable(s).
5. Solve the problem.
6. Interpret the solution of the inequality in terms of the context
of the problem.
You Try: The time has come for you to open a checking
account. A local bank is offering you a free checking account if
you maintain a minimum balance of $200. You already have a
savings account with this bank and you have $60 saved. You
decide to keep saving money until you have enough to open a
checking account, plus keep some money in savings. If you
deposit $15 a week into the savings account, what is the
minimum number of weeks it will take for you to be able to open
a checking account with at least $200 and still have $25 in your
savings account?
Warm Up
This year, Zachary has been babysitting his young cousins after
school for $70 a month. His uncle also gave him an extra bonus of
$100 for his excellent work. Since school started, Zachary has
earned more than $500. How many months ago did school start?
Write an inequality that represents the situation. Solve it and show
all your work.
Vocabulary and Key Concepts
• Exponential Equations: equations that have the variable in the
•
•
•
•
•
exponent
The general form of an exponential equation is
The base, b, must always be greater than 0
If the base is greater than 1, then the exponential equation
represents exponential growth
If the base is between 0 and 1, then the exponential equation
represents exponential decay
Another form of an exponential equation is
The pieces of the equation
• a: the initial value – look for words such as initial or
•
•
•
•
starting
r: rate of growth or decay
Use (1+r) for exponential growth problems
Use (1 – r) for exponential decay problems
Look for words such as double, triple, half, quarter –
which give the value for the base
Example 1: A population of mice quadruples every 6 months. If a
mouse nest started out with 2 mice, how many mice would there
be after 2 years? Write an equation and then use it to solve the
problem.
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known quantities
and variable(s).
5. Solve the problem.
6. Interpret the solution of the inequality in terms of the context of
the problem.
Example 2: In sporting tournaments, teams are eliminated after
they lose. The number of teams in the tournament decreases by
half with each round. If there are 16 teams left after 3 rounds,
how many teams started out in the tournament?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known
quantities and variable(s).
5. Solve the problem.
6. Interpret the solution of the inequality in terms of the context
of the problem.
Example 3: The population of a small town is increasing at a
rate of 4% per year. If there are currently about 6,000 residents,
about how many residents will there be in 5 years at this growth
rate?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known
quantities and variable(s).
5. Solve the problem.
6. Interpret the solution of the inequality in terms of the context
of the problem.
Example 4: You want to reduce the size of a picture to place in
a small frame. You aren’t sure what size to choose on the
photocopier, so you decide to reduce the picture by 15% each
time you scan it until you get it to the size you want. If the
picture was 10 inches long at the start, how long is it after 3
scans?
1. Read the problem.
2. Reread the scenario and make a list of the known quantities.
3. Read the statement again and look for the unknown or the
variable.
4. Create expressions and inequalities from the known
quantities and variable(s).
5. Solve the problem.
6. Interpret the solution of the inequality in terms of the context
of the problem.
You Try: On opposite sides of a major city two suburban
towns are experiencing population changes. One town, Town
A, is growing rapidly at 5% per year and has a current
population of 39,000. Town B has a declining population at a
rate of 2% per year. Its current population is 55,000.
Economists predict that in 5 years the populations of these
two towns will be about the same, but the residents of both
towns are in disbelief. The economists also claims that ten
years after that, Town A will double the size of Town B. Can
you verify the predictions based on the data given? Do you
think these predictions will come true?