Algebra 1 Unit 2A Notes: Reasoning with Linear Equations ...

[Pages:38]Algebra 1

Unit 2A: Equations & Inequalities

Notes

Name: ______________________ Block: __________ Teacher: _______________

Algebra 1

Unit 2A Notes: Reasoning with Linear

Equations and Inequalities

DISCLAIMER: We will be using this note packet for Unit 2A. You will be responsible for bringing this packet to class EVERYDAY. If you lose it, you will have to print another one yourself. An electronic copy of this packet can be found on my class blog.

1

Algebra 1

Unit 2A: Equations & Inequalities

Standard

MGSE9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions, quadratic, simple rational, and exponential functions (integer inputs only).

MGSE9-12.A.CED.2 Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MGSE9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as possible (i.e. a solution) or not possible (i.e. a non-solution) under the established constraints.

MGSE9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations.

MGSE9-12.A.REI.1 Using algebraic properties and the properties of real numbers, justify the steps of a simple, one-solution equation. Students should justify their own steps, or if given two or more steps of an equation, explain the progression from one step to the next using properties.

MGSE9-12.A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.

MGSE9-12.A.REI.5 Show and explain why the elimination method works to solve a system of two-variable equations.

MGSE9-12.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

MGSE9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

MGSE9-12.A.REI.11 Using graphs, tables, or successive approximations, show that the solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same.

MGSE9-12.A.REI.12 Graph the solution set to a linear inequality in two variables

Notes

Lesson

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Algebra 1

Unit 2A: Equations & Inequalities

Notes

Unit 2A:Equations & Inequalities

After completion of this unit, you will be able to...

Table of Contents

Learning Target #1: Creating and Solving Linear Equations ? Solve one, two, and multi-step equations (variables on both sides) ? Justify the steps for solving a linear equation ? Create and solve an equation from a context

Learning Target #2: Creating and Solving Linear Inequalities ? Solve and graph a linear inequality ? Create and solve an inequality from a context

Lesson

Day 1 ? Solve 1, 2 & MultiStep Equations

Day 2 ? Solving Equations with Fractions & Justifying Solving

Day 3 ? Graphing & Solving Inequalities

Page 4 7

10

Learning Target #3: Isolating a Variable ? Solve a literal equation (multiple variables) for a specified variable ? Use a Formula to Solve Problems

Learning Target #4: Creating and Solving Systems of Equations ? Identify the solution to a system from a graph or table ? Graph systems of equations ? Determine solutions to a system of equations ? Use a graphing calculator to solve a system of equations ? Use substitution & elimination to solve a system of equations ? Determine the best method for solving a system of equations ? Apply systems to real world contexts

Day 4 ? Creating Equations 13

& Inequalities

Day 5 ? Solving For a

15

Variable

Day 6 ? Graphing Systems 18

of Equations

Day 7 ? Solving Systems by 21

Substitution

Day 8 ? Solving Systems by 23

Elimination

Learning Target #5: Creating and Solving Systems of Inequalities ? Graph linear inequalities & systems of linear inequalities ? Create a linear inequality or system of inequalities from a graph ? Determine the solution to a linear inequality or system of inequalities ? Determine if a given solution is a solution to an inequality or system of inequalities ? Apply inequalities to real world contexts

Day 9 ? Real World

26

Applications of Systems

Day 10 ? More Real World 29

Applications

Day 11 ? Graphing Linear

32

Inequalities

Day 12 ? Graphing

36

Systems of Inequalities &

Applications

Monday August 19th

26th Day 3 ? Graphing & Solving Inequalities

2nd No School ? Labor Day

9th Day 11 ? Graphing Linear Inequalities

Tuesday 20th

27th Day 4 ? Creating Equations & Inequalities from a Context

3rd Day 7 ? Solving Systems

of Equations by Substitution

10th Day 12 ? Graphing

Systems of Inequalities

Wednesday 21st

28th Early Release Day Mixed Practice ? Solving Equations & Inequalities

4th Day 8 ? Solving Systems

of Equations by Elimination

11th Unit 2A Test Review

Thursday 22nd

Day 1 ? Solve 1, 2 & Multi-Step Equations

29th Day 5 ? Solving For a Variable Quiz on Creating & Solving Equations and Inequalities

5th Day 9 ? Real World

Applications Systems of Equations

12th Unit 2A Test

Friday 23rd

Day 2 ? Solving Equations with Fractions & Justifying

Solving 30th Day 6 ? Graphing Systems of Equations

6th Day 10 ? More Real World Applications Systems of Equations Quiz

13th

3

Algebra 1

Unit 2A: Equations & Inequalities Day 1 ? Solving One & Two Step Equations

Notes

Standard(s): MGSE9-12.A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.

Expression: ? A mathematical "phrase" composed of terms, coefficients, and variables that stands for a single number, such as 3x + 1 or x2 ? 1. ? We use Properties of Operations to simplify algebraic expressions. Expressions do NOT contain equal signs.

Equation: ? A mathematical "sentence" that says two expressions are equal to each other such as 3x + 1 = 5. ? We use Properties of Equality (inverse operations) to solve algebraic equations. ? Equations contain equal signs.

When solving equations, you must perform inverse operations, which means you have to perform the operation opposite of what you see. You must also remember the operation you perform on one side of the equation must be performed to the other side.

Informal

Operation

Inverse

Addition

Subtraction

Multiplication

Division

Property

Addition Property of Equality

Subtraction Property of Equality

Multiplication Property of Equality

Division Property of Equality

Formal

General Example

If a = b, then a + c = b + c

If a = b, then a ? c = b - c

If a = b, then ac = bc

If a = b, then

Example 1 If x ? 4 = 8, then x = 12 If x + 5 = 7, then x = 2

If , then x = 18

If 2x = 10, then x = 5

Additive Inverse

Multiplicative Inverse (Reciprocal)

A number plus its inverse equals 0.

A number times its reciprocal equals 1.

a + -a = 0 a =1

7 + -7 = 0 3 =1

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Algebra 1

Unit 2A: Equations & Inequalities

Notes

Solving One Step Equations Practice

Practice: Solve each equation.

1.

x ? 4 = 3

Operation You See: _______________

Inverse Operation: _______________

2.

y + 4 = 3

3.

s =9

3

4.

6p = 12

Operation You See: _______________ Operation You See: _______________ Operation You See: _______________

Inverse Operation: _______________ Inverse Operation: _______________ Inverse Operation: _______________

Practice: Solve each equation on your own.

a.

x ? 6 = 10

b.

-5d = 25

c.

8 + m = -4

d.

x =1

7

e.

y ? (-9) = 2

f.

Solving Two Step Equations

When solving equations with more than one step, you still want to think about how you can "undo" the operations you see.

Practice: Solve each equation, showing all steps, for each variable.

1. 3x - 4 = 14

2. 2x + 4 = 10

3. 7 ? 3y = 22

4. 0.5m ? 1 = 8

5. -6 + = -5

6. x - 8 = -5 4

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Algebra 1

Unit 2A: Equations & Inequalities Solving Multi-Step Equations

Notes

Multi-step equations mean you might have to add, subtract, multiply, or divide all in one problem to isolate the variable. When solving multi-step equations, you are using inverse operations, which is like doing PEMDAS in reverse order.

Multi - Step Equations with Combining Like Terms

Practice: Solve each equation, showing all steps, for each variable.

a. -5n + 6n + 15 ? 3n = -3

b. 3x + 12x ? 20 = 25

c. -2x + 4x ? 12 = 40

Multi - Step Equations with the Distributive Property

Practice: Solve each equation, showing all steps, for each variable.

a. 2(n + 5) = -2

b. 4(2x ? 7) + 5 = -39

c. 6x ? (3x + 8) = 16

Multi ? Step Equations with Variables on Both Sides

Practice: Solve each equation, showing all steps, for each variable

a. 5p ? 14 = 8p + 4

b. 8x ? 1 = 23 ? 4x

c. 5x + 34 = -2(1 ? 7x)

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Algebra 1

Unit 2A: Equations & Inequalities Day 2 - Equations with Fractions and Decimals

Notes

Standard(s): MGSE9-12.A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.

MGSE9-12.A.REI.1 Using algebraic properties and the properties of real numbers, justify the steps of a simple, one-solution equation. Students should justify their own steps, or if given two or more steps of an equation, explain the progression from one step to the next using properties.

When you solve equations with decimals, you solve them as if you would an equation without decimals.

1. 3.5x ? 37.9 = .2x

2. 14.7 + 2.3x = 4.06

3. -1.6 ? 0.9w = 11.6 + 2.4w

Equations with Fractions

When solving equations with fractions, you want to find a way to eliminate the fraction.

To eliminate the fraction, multiply by a Common Denominator

1. - 2 m = 10 3

2.

3x = 6

4

3.

- 3 x -1= 8

2

4.

2m + 5 = 12

3

1.

w + 1 = 6w -1

77

2.

x + 2x = 5

63

3. x + 3 - x = 5 82

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Algebra 1

Unit 2A: Equations & Inequalities

Special Types of Solutions

Solve the following equations. What do you notice about the solutions?

a. 2x ? 7 + 3x = 4x + 2

b. 3(x ? 5) + 11 = x + 2(x + 5)

Notes

c. 3x + 7 = 5x + 2(3 ? x) + 1

Justifying the Solving of Equations

Property Commutative Property of

Addition Associative Property of

Addition

Additive Identity

Additive Inverse

Commutative Property of Multiplication

Associative Property of Multiplication

Multiplicative Identity Multiplicative Inverse

(Reciprocal) Zero Property of

Multiplication

Distributive Property

Properties of Addition Operations

What It Means

General Example

Rearrange the order and the sum will stay the same.

Change the order of the grouping and the sum will

stay the same. Zero added to any number will equal that

number. A number plus its inverse

equals 0.

a + b = b + a (a + b) + c = a + (b + c)

a + 0 = a a + -a = 0

Properties of Multiplication Operations

Rearrange the order and the product will stay the

same. Change the order of the grouping and the product

will stay the same. One times any number

equals that number.

a b = b a (a b) c = a (b c)

a 1 = a

A number times its reciprocal equals 1.

a =1

Any number times 0 will always equal 0.

a 0 = 0

Multiply a number to every term within a quantity (parenthesis).

a(b + c) = ab + ac

Example 1 2 + 4 = 4 + 2

(4 + 6) + 1 = 4 + (6 + 1)

4 + 0 = 4 7 + -7 = 0

5 2 = 2 5

(3 4) 2 = 3 (4 2) 8 1 = 8 3 =1 7 0 = 0

4(x + 5) = 4x + 4(5) = 4x + 20

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