Topic 1

Mrs. Daniel- Algebra 1

Table of Contents• 1.1: Solving Equations

• 2.1: Modeling with Expressions

• 2.2: Creating & Solving Equations

• 2.3: Solving for Variable

• 2.4: Creating & Solving Inequalities

•2.5: Creating & Solving Compound Inequalities

Lesson 1.1Solving Equations

Essential Question: How do you solve an equation in one

variable?

How to solve Multi-Step Equations1. Distribute.

Bonus Step: Multiple by reciprocal/LCM, if fractions.

2. Combine Like Terms (ONLY on same side of equal sign).

3. Use the inverse operation to move numbers to the right.

4. Use the inverse operation to move variables to the left.

5. Divide.

Example

6(4 + 3x) - 10x = 30 - 2x

Example

1

5x – 8 = 4 + 2x

Let’s Practice #1

4(5 + 2x) + 7x = -5

Let’s Practice #2

3

2𝑥 + 6 = −

7

2+ 5𝑥

Justify Each Step.

Step # Math Justification

Given 3x – 2 = 6

1 3x – 2 + 2 = 6 +2

2 3x = 8 Combine Like Terms

33𝑥

3= 8

3

4 x = 8

3Simplify.

Let’s Practice #3: Justify Each Step.

Step # Math Justification

Given1

2𝑧 + 4 = 10

11

2𝑧 + 4 − 4 = 10 − 4

21

2𝑧 = 6

3 2 ∙1

2𝑧 = 6 ∙ 2

4 z = 12 Simplify.

Let’s Practice #4: Justify Each Step.

Step # Math Justification

Given 5𝑥 − 10 = 20

1

2

3

4 x = 6 Simplify.

Let’s Practice #5:

6. An ostrich that is 108 inches tall is 20 inches taller than 4 times the height of a kiwi. What is the height of a kiwi in inches?

7. An emu that measures 60 inches in height is 70 inches less than 5 times the height of a kakapo. What is the height of the kakapo in inches?

Lesson 2.1: Modeling with

Expressions

Essential Question: How do you interpret algebraic expressions in terms of their

context?

Vocab

Expression: a mathematical phrase that contains operations, numbers, and/or variables.

The terms of an expression are the parts that are being added.

Coefficient: is the numerical factor of a variable term.

1. Curtis is buying supplies for his school. He buys p packages of crayons at $1.49 per package and q packages of markers at $3.49 per package. What does the expression 1.49p + 3.49q represent?

2. Sandi buys 5 fewer packages of pencils than p packages of pens. Pencils cost $2.25 per package and pens cost $3 per package. What does the expression 3p + 2.25(p – 5) represent?

Write an algebraic expression:

3. The price of an item plus 6% sales tax

4. The price of a car plus 8.5% sales tax

5. The number of gallons of water in a tank, that already has 300 gallons in it, after being filled at 35 gallons per minute for m minutes

Lesson 2.2:Creating & Solving

Equations

Essential Question: How do you use an equation to model and solve a

real-world problem?

Write and solve an equation.

1. Aaron and Alice are bowling. Alice’s score is twice the difference of Aaron's score and 5. The sum of their scores is 320. Find each student’s bowling score.

Write and solve an equation.

2. Mari, Carlos and Amanda collect stamps. Carlos has five more stamps than Mari and Amanda has three times as many stamps as Carlos. Altogether, they have 100 stamps. Find the number of stamps each person has.

Write and solve an equation.3. A rectangular garden is fenced on all sides with 256 feet of fencing. The garden is 8 feet longer than it is wide. Find the length and width of the garden.

Write and solve an equation.4. Janine has job offers at 2 companies. One company offers a starting salary of $28,000 with a raise of $3000 each year. The other company offers a starting salary of $36,000 with a raise of $2000. In how many years would Janine’s salary be same with both companies? What will the salary be?

Write and solve an equation.5. Claire bought just enough fencing to enclose either a rectangular garden or a triangular garden, as show. The two gardens have the same perimeter. How many feet of fencing did she buy?

Write and solve an equation.6. Lisa is 10 centimeters taller than her friend Ian. Ian is 14 centimeters taller than Jim. Every month, their heights increase by 2 centimeters. In 7 months, the sums of Ian’s and Jim’s heights will be 170 centimeters more than Lisa’s height. How tall is Ian now?

7. Dave and Steve are 700 meters apart from each other on a straight path. Both start walking toward each other and meet each other after 7 minutes. Dave covers a distance of x meters in one minute, which is 20 meters less than the distance Steve covers in a minute. Which of these equations can be used to find the rate at which Steve and Dave walk?

Lesson 2.3:Solving for a Variable

Essential Question:

How do you rewrite formulas and literal questions?

Vocab:

Literal Equations are equations that contain 2 or more variables.

Solve. 1. Solve for l; V = lwh

2. Solve for h; V = Bh

3. Solve for m; y =mx + b

1. The formula for density is D = 𝑚

𝑉. Lead has a very

high density of 11,340kg/m3. A sinker on a fishing line is made of lead and has a volume of 0.000015m3. What is the mass of the sinker? (Calculator = YES)

2. The formula for density is D = 𝑚

𝑉. Plastic foam has a

very low density of 75 kg/m3. The design for a life preserver requires 0.3 kilograms of plastic foam to provide proper buoyancy. What is the volume of the plastic foam required?

3. For altitudes up to 36,000 feet, the relationship between ground temperature and atmospheric temperature can be described by the formula t = -0.0035a + g, where

t is atmospheric temperature in degrees Fahrenheit

a is the altitude, in feet, at which the atmospheric temperature is measured

g is the ground temperature in degrees Fahrenheit.

Determine the altitude in feet when t is -27.5°F and g is 60°F.

Let’s Practice # 4

Interest Formula

I = prtI = Interest paid

p = principal (starting amount)

r = rate (as a decimal)

t = time

5. Find the number of years used in the calculation of a $1000 loan at an interest rate of 5% with interest totaling $600.

6. Determine the interest rate for a $2000 loan that will be paid off in 4 years with interest totaling $640.

Lesson 2.4:Creating & Solving

Inequalities

Essential Question: How do you write and solve an inequality that represents a real-world

situation?

Special Warning:

If you multiply or divide by a negative; change the direction of the inequality symbol!!!

1. Trina is buying 12 shirts for the drama club. She will choose a style for the blank shirts and then pays an additional charge of $2.75 for each shirt to have the club logo. If Trina cannot spend more than $99, how much can she spend on each blank shirt? Write an solve an inequality to find the cost of each blank shirt.

2. Sergio needs to buy gifts for 8 friends. He wants to give the same gift to all his friends and he plans to have the gifts wrapped for an additional charge of $1.50 each. If Sergio spends a total of at least $70, he will receive free shipping on his order. Write and solve an inequality to determine how much Sergio needs to sped on EACH gift to receive free shipping.

3. 4.

5. The Daily Info charges a fee of $650 plus $80 per week to run an ad. The People’s Paper charges $145 per week. For how many weeks must an ad run for the total cost at the Daily Info to be less expensive than the cost at the People’s Paper? Let w be the number of weeks the ad runs in the paper.

6. The Home Cleaning Company charge $312 to pressure cleaning the side of a house plus $12 for each window. Power Clean charges $36 per window, and the price includes pressure cleaning the siding. How many windows must a house have to make the total cost from The Home Cleaning Company less expensive than Power Clean? Let w be the number of windows.

7. Alfonso makes $8 per hour working at a movie theater and $12 per hour working at a restaurant. Next wee, Alfonso is scheduled to work 6 hours at the movie theater. Which of the following inequalities represent the amount of hours (h) that Afonso needs to work at the restaurant next week to earn at least $144 between his two jobs?

8. The school band will sell pizzas to raise money for new uniforms. The pizza supplier charges $100 plus $4 per pizza. The band members sell the pizzas for $7 each. Write and solve an inequality to find how many pizzas the bank members will have to sell to make a profit.

2.5: Creating & Solving Compound Inequalities

Essential Question: How can you solve a compound inequality and

graph the solution as a set?

What is a Compound Inequality?

The combination of two or more simple inequalities creates a compound inequality.

The graphs can involve an intersection, AND or overlapping regions.

The graphs can involve a union, OR or diverging regions.

How to Solve a Compound Inequalities

Step 1: Separate into separate inequalities.

Step 2: Solve.

Step 3: Combine solutions.

Step 4: Graph each solution on the same number line.

Example

- 3 ≤ m – 4 < -1

Let’s Practice #1a. 3 < 2p 3 12

b. 6b 1 41 or 2b + 1 11

Let’s Practice #2

3 >11+𝑘

4≥ -3

Let’s Practice #3

Let’s Practice #4

Let’s Practice #5

Write a compound inequality to represent the indicated quality control level and graph.

The recommended alkalinity level for swimming pool water is between 80 and 120 parts per million, inclusive.