Mrs. Bondi - Algebra 1 2105/2106/2716
Unit 2 Class Notes
Unit 2:
Equations &
Inequalities
Lesson Topics:
Lesson 1 Solving One-Step Equations (PH text 2.1, 2.2)
Lesson 2 Solving Two-Step Equations (PH text 2.2)
Lesson 3 Solving Real-Life Problems (PH text 2.1-2.2)
Lesson 4 Solving Multi-Step Equations (PH text 2.3)
Lesson 5 Solving Equations with Variables on Both Sides (PH text 2.4)
Lesson 6 Literal Equations and Formulas (PH text 2.5)
Lesson 7Consecutive Integer Problems (PH text -not available)
Lesson 8 Ratios, Rates, and Conversions (PH text 2.6)
Lesson 9 Solving Proportions (PH text 2.7)
Lesson 10 Proportions and Similar Figures (PH text 2.8)
Lesson 11 Percents (PH text 2.9)
Lesson 12 Change Expressed as a Percent (PH text 2.10)
Lesson 13: Pythagorean Theorem (PH text 10.1)
Lesson 14 Inequalities and Their Graphs (PH text 3.1)
Lesson 15 Solving Inequalities (PH text 3.2-3)
Lesson 16 Solving Multi-Step Inequalities (PH text 3.4)
Lesson 17 Compound Inequalities (PH text 3.6)
Lesson 18 Absolute Value Equations and Inequalities (PH text 3.7)
Lesson 1 Solving One-Step Equations (PH text 2.1, 2.2)
Objectives: to identify and explain the first four properties of equality
to solve one-step equations
to use equations to solve real-world problems
Addition Property of Equality
For any real numbers a, b and c, if a = b , then a + c = b + c .
Subtraction Property of Equality
For any real numbers a, b and c, if a = b , then a – c = b – c .
Multiplication Property of Equality
For any real numbers a, b and c, if a = b , then a ∙ c = b ∙ c .
Division Property of Equality
For any real numbers a, b and c, if a = b , then .
Inverse operations – operations that “undo” one another
Solution to an equation – any value(s) that make the equation true.
We can use our understanding of inverse operations and the properties of equality to find a solution to a given equation.
Examples:
1)2)
3)4)
5)6)
Reciprocal (multiplicative inverse) – for any nonzero , the reciprocal is
the product of any nonzero number and its reciprocal is one (1) – zero does not have a reciprocal
Multiplying by a Reciprocal:
If the coefficient of a variable is a fraction, multiply each side by its reciprocal.
7)8)
9)10)
HW: p. 85 #20-48 multiples of 4, 55-59
Reminder: Homework must be done on a separate sheet of paper. The assignment must be written at the top. The problem must be written out, and all work must be shown. The answer must be boxed/circled.
Lesson 2 Solving Two-Step Equations (PH text 2.2)
Objective: to solve two-step equations
When we solve a problem using order of operations, we multiply or divide before we add or subtract. When we are solving for a variable, we need to undo that process, so we need to use order of operations in reverse.
Basic Math Algebra
4(2) + 3 = 4(x) + 3 = 11
– 3 – 3
8 + 3 =4x = 8
11
x = 2
Use inverse operations to isolate the variable.
1st - Move any number added or subtracted to the variable.
2nd - Move any number multiplied by or dividing the variable.
Examples:
1)2)
3)4)
5)6)
7)8)
If the variable is part of an expression in the numerator of a fraction, multiply each side by the denominator.
9)10)
11)
HW: p. 91 #19-22, 27-48 multiples of 3
Reminder: Homework must be done on a separate sheet of paper. The assignment must be written at the top. The problem must be written out, and all work must be shown. The answer must be boxed/circled.
Lesson 3 Solving Real-Life Problems (PH text 2.1-2.2)
Objectives:To translate a word statement into an equation
We can use our ability to translate verbal expressions into algebraic expressions to solve real world problems. See - math has a purpose!
Step 1 – Read through the whole problem.
Step 2 –Identify the thing to be found.
Step 3 –Identify the variable.
Step 4 – Translate the verbal expressions into an algebraic equation.
Step 5 – Reread the problem to make sure the verbal and algebraic expressions match.
Step 6 – Solve the equation.
Step 7 – Check to be sure the answer makes sense.
Step 8 – Write out the answer to the question completely.
Examples:
1)James, who currently weighs 376 lb, wishes to compete as a middle-weight in a sumo wrestling tournament next month. To enter the tournament, he must weigh no more than 360 lb. How much weight must he lose before next month? Write an equation and solve.
2) Helen is planning to go to the grocery store to buy eggs for her bagel shop. If each carton of eggs contains 12 eggs, how many cartons must she buy to obtain 450 eggs? Write an equation and solve.
3)Frieda, a cab driver, rents her cab from Super Taxi for $50 a day. If she makes an average of $6 each time she picks up a passenger, how many passengers must she pick up to make $75 profit?
4) Kiera and her friend plan to hike and camp in the state park. If the yurt can be rented for $18 per night, plus a cleaning fee of $12 for the visit, how many nights did they stay if their total cost was $66?
HW: p. 85 #50, 51, 54, 70; p. 91 #23-25, 58-60
Follow correct homework form. You only need to write the algebraic expression, not the verbal.
Lesson 4a Solving Multi-Step Equations (PH text 2.3)
Objective: to solve multi-step equations that involve combining like terms and the distribute property
When the variable occurs in two or more terms,
simplify the equation before solving it by combining like terms.
Class Practice:
Solve each equation.
1)2m + 3m + 4 = 142)8x – 4x = 24 – 12 3)18 – 5x + x – 1 = 5
Distributive Property:
For all real numbers a, b, and c:
a(b + c) = ab + ac(b + c)a = ba + ca
a(b – c) = ab – ac(b – c)a = ba – ca
Class Practice:
Solve each equation.
7)8)9)
10)11) 12)
HW: p. 97 #9, 16-20, 27-29, 55-57
Lesson 4b Solving Multi-Step Equations (PH text 2.3)
Objective: to solve equations involving rational numbers
When equations contain fractions or decimals, it is often easier to clear them from the equation using properties of equality.
Multiplying by a power of ten:
If any of the numbers in the equation is a decimal, you can multiply each side by a power of ten that will make all of the numbers integers.
1)2)
Multiplying by a Common Denominator:
If the variable is part of an expression in the numerator of a fraction, multiply each side by the denominator.
If the coefficient of the variable is a fraction, multiply each side by the least common denominator.
3)4)same as
Class Practice:
5)6)7)
8)9)10)
11)12)
Lesson 5 Solving Equations with Variables on Both Sides (PH text 2.4)
Objective: to solve equations with variables on both sides
If there is a variable on both sides of the equation,
- Add or subtract to move one variable to the other side.
- Combine like terms.
- Solve the equation.
Class Practice:
1)2)3)
4)5)6)
Special Case: No Solution => 3x – 2 = 6 + 3x => no value of the variable can make the equation true
Special Case: Identity => 3x + 6 = 6 + 3x => true for every possible value of the variable
7)Chris and Maggie decide to meet after school at a nearby music store. Chris walks to the store at a
speed of 2 mi/h. Maggie bicycles along the same path at 7 mi/h. Maggie gets to the store 30 minutes
before Chris. How long did it take Maggie to get to the store?
HW: p.105 #17-22, 39-44
Lesson 6 Literal Equations and Formulas (PH text 2.5)
Objective: to rewrite and use literal equations and formulas
Literal equation – an equation that has two or more variables
Example:
Bridget is hosting a party where she will serve slices of pizza and hoagies. Padrino’s Pizza will sell her pizzas for $10 each and hoagies for $5 each. She wants to spend no more than $80 for the food.
How many hoagies can she buy if she buys 3 pizzas? 6 pizzas?
Equation: 10p + 5h = 80We want to know how many hoagies ….. h = ?
By first rearranging the equation, solving for h, we save ourselves work.
Class Practice:
Formulas state the relationship among quantities. Formulas are frequently manipulated to solve for different unknowns. Formulas are special types of literal equations.
Example:
Use the simple interest formula, I = prt to solve word problems involving yearly interest.
I= prt where I = interestp = principle
r = rate per yeart = time in years
You invest $1500 for 2 years. You earn $90 in simple interest. What is the annual rate of interest?
Class Practice:
- What is the width of a rectangle with length 14 cm and area 161 cm2?
- What is the radius of a circle with circumference 13 ft?
- A rectangle has perimeter 182 in. and length 52 in. What is the width?
- A triangle has base 7 m and area 17.5 m2. What is the height?
The distance formula tells us that the distance traveled is equal to the product of the rate of speed and the amount of time traveled. d=rt.
5.Find the time it will take me to drive to my aunt’s house in Ocean Park, Maine (425 miles from home) if I average 65 mph.
HW: p.112 #18-38 even, 45
Lesson 7 Finding Consecutive Integers(PH text – not in there)
Objective: to solve word problems involving consecutive integers
Consecutive Integers:
A)Name the next three consecutive integers. -1, ____, ____, ____
What is their sum? _____
Represent these integers with algebraic expressions.____, ______, ______, ______
What is their sum? ______
B)Name the next three consecutive odd integers. -1, ____, ____, ____
What is their sum? _____
Represent these integers with algebraic expressions.____, ______, ______, ______
What is their sum? ______
HW: Practice Exercises #1-6, 13-16
Lesson 8 Ratios, Rates, and Conversions (PH text 2.6)
Objective: to find ratios and rates and to convert units and rates
Ratio – a comparison of two numbers by division – often looks like a fraction - can be written in three forms => a to b, a:b, (or a/b)
Rate – a comparison of two quantities measured in different units – often looks like a fraction
Unit Rate – a rate with a denominator of 1 (one) unit
Conversion Factor – a ratio of two equivalent measures in different units (so always = 1) - allows you to convert from one unit to another through multiplication (conversion table – p.801)
Unit Analysis (dimensional analysis) – including the units for each quantity in the calculations to help determine the unit of the answer
You can write ratios and find unit rates to compare quantities. You can also convert units and rates to solve problems.
HW: p. 121 #1, 9-14, 23-28
Lesson 9 Solving Proportions (PH text 2.7)
Objective: to solve and apply proportions
Proportion – a statement that two ratios are equal --- example:
Cross Products Property:
The cross products of a true proportion are equal.
Use cross products to decide if each proportion is true.
Class Practice:
1.2.3.
4.5.6.
Sometimes there is more than one term in part of the proportion. The Cross Products Property still applies; it just results in a multi-step problem.
7.8.9.
10. My computer takes 10 minutes to download a 45-min. TV show. How long will it take the same computer to download a 2-hour movie? Write a proportion to solve.
HW: p.127 #18-32even, 34-36, 51
Lesson 10 Proportions and Similar Figures (PH text 2.8)
Objective: to find missing lengths in similar figures
to use similar figures when measuring indirectly
Similar figures are proportional. They have the same shape, but not the same size. Their corresponding angles are equal, and their corresponding sides have proportionate lengths.
10
6
35
84
Class Practice:
Scale drawings or scale models help us evaluate something that is very large by having a proportionately smaller representation of the original. A proportion can be used to find a real-life measurement from the drawing using the scale.
Scale - a ratio representing the size of an illustration or reproduction, in relation to the object it represents – usually for a map or a model – example: ½ inch: 10 miles
Class Practice:
HW: p.134 #8-22 even, 23, 25-26
Lesson 11 Percents (PH text 2.9)
Objective: to solve percent problems using proportions.
To solve percent problems using the percent equation.
Percent Proportion:
You can represent “a is p percent of b” using a proportion.
b represents the base and a represents a part of base b
Example:
Your quiz score is 23 out of 25, but you want to know the percentage.
base = Plug the values. Solve using the Cross Products Property.
part of the base =
percent =
Helpful Hint:
“is” means ______“of” means ______“what” means ______
Plug the values. Solve using the Cross Products Property.
Write an equation to solve each problem.
What is 60% of 8?8 is 25% of what number?What percent of 30 is 5?
Three is 15% of what number?What percent of 44 is 11?What is 75% of 80?
HW: p.141 #10-42 even
Lesson 12 Change Expressed as a Percent (PH text 2.10)
Objective: to find percent change
to solve word problems involving percent of increase or decrease.
Look at how much something goes up or down, and compare it to what it originally was.
Percent of Change: the percent an amount changes from its original amount
Percent of Change =
Percent of Increase: when a value increases from its original amount
Percent of Decrease: when a value decreases from its original amount
Find the percent of change for each problem and describe it as change of increase or decrease.
Round to the nearest percent.
1.original amount $12, new amount $9
2.original amount 19 in., new amount 25 in.
3.original amount 180 lb, new amount 150 lb
4.original amount 5ft., new amount 5ft
Relative Error: the ratio of the absolute value of the difference of measured or estimated value and an actual value compared to the actual value.
relative error =
Percent Error is when the relative error is expressed as a percent.
5.I estimated it would take us 10 hours to clean out the garage. It actually took us 11 ½ hours. What was the percent of error in my estimate?
6.We estimated that we would drive about 600 miles during our visit to Colorado. What was our percentage of error in our estimate if we actually drove 721 miles?
HW: p.148 #1-2, 4, 8-18 even, 19, 26-30 even, 37
Lesson 13: Pythagorean Theorem(PH text 10.1)
Objective: Students will be able to use the Pythagorean Theorem and it’s Converse.
Pythagorean Theorem
In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
Converse of the Pythagorean Theorem
If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
Vocabulary:
Hypotenuse – 1) ______
2) ______
Legs - ______
Pythagorean Triple – when the lengths of the sides of a right triangle are ______.
1.Find the length of the missing sides.
3a)b) c)
x = ______x = ______x = ______
2. An 11 foot ladder is leaning against a tree. The base of the ladder is 4 feet from the tree. To the nearest foot, how high up the tree does the ladder reach?
3. A brick walkway forms the diagonal of a square playground. The walkway is 24 meters long. To the nearest tenth of a meter, how long is a side of the playground?
4.Tell if the following sets of numbers are Pythagorean triples.
a)5, 8, 10b) 4, 40, 41c)5, 12, 13
Practice:
HW: p.602 #15-25, 29, 37
Unit 2, Part 1 Test Review:
Lesson 1:Solving One-Step Equations (PH text 2.1, 2.2)
Lesson 2: Solving Two-Step Equations (PH text 2.2)
Lesson 3: Solving Real-Life Problems (PH text 2.1-2.2)
Lesson 4: Solving Multi-Step Equations (PH text 2.3)
Lesson 5: Solving Equations with Variables on Both Sides (PH text 2.4)
Lesson 6: Ratios, Rates, and Conversions (PH text 2.6)
Lesson 7: Solving Proportions (PH text 2.7)
Lesson 8: Proportions and Similar Figures (PH text 2.8)
Lesson 9: Literal Equations and Formulas (PH text 2.5, 6.4)
Lesson 10: Percents (PH text 2.9)
Lesson 11: Change Expressed as a Percent (PH text 2.10)
Lesson 12: Finding Consecutive Integers(PH text – not in there)
Lesson 13: Pythagorean Theorem (PH text 10.1)
Practice in Prentice Hall Text:
Ch.2, all lessons (odd answers in back of book); 10.1
Review – p. 152-156; 642 #6-22 (all answers in back of book)
Practice test – p.157; 645 #1-7
Cumulative Test preparation – p. 158-160
HW after test: p.161 (Get ready for ch.3!)
Lesson 14 Inequalities and Their Graphs (PH text 3.1)
Objective: to graph inequalities with one variable
Equation – a mathematical sentence that uses an ______sign to show that the two expressions have the ______value
Inequality – a mathematical sentence that uses an inequality symbol/sign to ______the values of two expressions
Use a representative arrow on a number line to visually indicate the values that make the inequality true.
Equality SymbolsTermGraphing Symbol
=equal to
Inequality SymbolsTermGraphing Symbol
less than
greater than
≤less than or equal to
≥greater than or equal to
≠not equal to
Be sure the variable is on the left side of the inequality before graphing. (It will be far easier!)
Graph each equality or inequality.
x = 1x ≠ 1
x ≤ 1x < 1
x ≥ 1x > 1
0 ≤ x ≤ 20 < x < 2
A solution to an inequality is any value that makes the inequality true. Every value darkened on your graph is a solution to the inequality.
Examples:
1)Determine whether each number is a solution of the given inequality.
2x +4 20a. 2b. 10
Graph each inequality.
2. n ≥ 53. j >–4
4. k ≤105. m < -3/2
Write an inequality for each graph.
6. 7.
Define a variable and write an inequality to model each situation.
8.Nomorethan10peoplemayusethetreadmillsatanytimeinthegym.
9.Totrainforamarathon,arunnerdecidesthatshemustrunatleast12miles each day.
Practice:
HW: p.168 #8-42 even, 65, 67, 75 (show fraction work)
Lesson 15a Solving Inequalities (PH text 3.2)
Objective: to solve and graph inequalities with one variable using addition or subtraction
Solution of the Inequality – any value(s) of the variable that makes the inequality true
Example: For x > 10, the solution is all numbers greater than 10.
To solve an inequality, follow the same process used to solve equalities.
When adding and subtracting, the process works exactly the same as solving an equality.
Example:x + 7 = 6x + 7 < 6
–7–7–7–7
x = –1x < –1
Set Notation: {x: x = –1}{x: x–1}
Set Builder Notation – demonstrates parts needed to build desired set – {x: x < -1}
Remember, it is easiest if you are sure the variable is on the left side of the inequality before graphing.
Examples: Solve and graph each inequality. Write your solution in set notation.
1)2)
3)4)
5)6)
7)A family earns at most $2500 a month. The family’s expenses are $2000. Write and solve an inequality to find the possible amounts of money the family could deposit into a savings account each month.
Practice: Write your solution in set notation.
HW: p.174 #12-44 multiples of 4, 70, 71, 81-82
Lesson 15b Solving Inequalities (PH text 3.3)
Objective: to solve and graph inequalities with one variable using multiplication or division
To solve an inequality, follow the same process used to solve equalities.
When multiplying and dividing by a positive number, the process works exactly the same as with an equation.
Example:7x = 427x42
7777
x = 6x < 6
BUT … When multiplying or dividing by a negative number, REVERSE THE SIGN of the inequality.
Example:–7x = 42-7x42
–7–7–7–7When multiplying or dividing by a negative number
x = –6x –6REVERSE THE SIGN of the inequality