1.4 Solving Equations Expectations · 1.6 Solving Absolute Value Equations and Inequalities A: Solving Absolute Value Equations Perhaps you remember from a previous math class the

Algebra 2 Chapter 1 Notes 1.4 Solving Equations

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1.4 Solving Equations Topics: Solving Equations Translating Words into Algebra Solving Word Problems

A: Solving One-Variable Equations The equations below are easy types of equations that you solved often in Algebra I.

One Step x + 6 = 9 -3y = 15

Two Step 2 3 6a In Algebra II, you will face some equations that are bit more challenging to solve. Example 1: Solve each of the following. Show your work!

1a. 6 5 7 9x x 1b. 5 3 2

6 8 3v

Expectations:

I want to see for any problem:

The original problem

Any key steps in getting to your

solution- “the work”

Clearly stated solution

Answers:

Should use original variable if

applicable x = 2 or y = 5, etc.

FRACTIONS should always be

reduced to lowest terms.

DECIMALS only if they are

terminating and you write the entire

thing… never round unless the

directions say so.

*Get like denominators OR use a scientific calculator with a fraction button*


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1c. 5(6 4 ) 21y y 1d. 53 = 3(y − 2) − (3y – 1)

1e. 1

1 22

x x 1f.

B: Translating Algebra to Words and back (review lesson 1.3)

Example 2: Write an algebraic expression to represent each verbal expression. 2a. three less than a number 2b. six times the cube of a number is the quotient of the same number and 81 2c. the square of a number decreased by the product of the same number and five

2d. twice the difference of a number and six is equal to 24

TIPS:

“A number” means the

variable… pick one! I like

“x” and “n”, but any letter

will do!

Try to pick out words that

mean operators (+, ×, ÷, −)

“is” means “=”

Use ( ) to group a “sum” (etc)

together.


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Example 3: Write a verbal expression to represent each equation. 3a. 6 = − 5 + x 3b. 7y − 2 = 19

3c. 3(2 1)5

mm

C: Word Problems Procedure for solving word problem

Solve each of the following story problems. Be sure to name a variable, show your equation and your work, and answer the question being asked in words. 1. a) The sum of three consecutive integers is 78. What are the integers? b) How would you change the variables for “Three consecutive odd integers”?

1. Relax. Read through the problem at least twice.

2. Identify a variable. Call it whatever you want, x, n, etc. and tell what it

means/stands for (example: let x = the 5th test score)

3. Write an equation that contains that variable by translating the English words

into Math symbols and numbers

4. Solve the equation.

5. Answer the question asked and make sure your answer makes sense

TIPS:

Read the expression out

loud… that is what you

should write down.

Write “a number” for any

variable

A group in parenthesis is

usually a “sum” “difference”

“product” or “quotient”

You must SPELL the first

word of any sentence even if

it is a number. After that,

you may use numerals.


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2. Sam’s scores on his first four tests are 84, 75, 86, and 92.

What must he score on the fifth test to average 88%? 3. Tickets to A Midsummer Night’s Dream cost $10.50 for adults and $7.50 for students. Mrs. Smith

ordered $192 worth of tickets for a field trip for her English class. If a total of 24 tickets were ordered, how many of each type of ticket did she order?

4. The length of a rectangle is twice as long as the width. Find the dimensions of the rectangle if

the perimeter is 360 ft. 5. The sides of a triangle are in a ratio of 4:5:6 . The perimeter of the triangle is 52.5 inches. Find

the lengths of each side.

Algebra 2 Chapter 1 Notes 1.5 Solving Inequalities

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1.5 Solving Inequalities

A: Writing, Solving, and Graphing Inequalities Inequalities represent values that may not necessarily be equal. Example: What inequality represents: “Fourteen minus 5 times a number is no more than 20” ?

You may recall from Algebra 1, that to graph an inequality in one variable, you use a closed or open dot on the number and then shade to the left or right.

Solving inequalities is the same as solving equations EXCEPT if you multiply or divide by a negative number, you have to FLIP the inequality symbol. Solve the example from above, then graph the solution. Examples 1: Solve each inequality. Graph the solution. 1a. 1b. 1c.

Inequality Summary greater than

less than

greater than or equal

less than or equal


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1d. ** 1e.**

B: Solving Compound Inequalities There are two types of compound inequalities: The “AND” and the “OR”

Type 1: “AND” Compound Inequalities “A” and “B” means the overlapping values common to A and B.

Example 2: Solve each compound inequality. Graph the solution. 2a. 3 12x and 8 16x

A B


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2b. 3 4 16x and 2 1 13x 2c. 13 2 7 17x

2d. 19 3 2 10y 2e. 5 15x and 6 8x

Type 2: “OR” Compound Inequalities “A” or “B” means the all the area covered by A and all area covered by B

A B


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Example 3: Solve each compound inequality. Graph the solution.

3a. 2 3 4 3y or y 3b. 4 1 9d or 2 5 11d

3c. 3 9x or 2 10x Example 4: 4a. A farmer wants to make a rectangular pen using no more than 400 ft. of fencing. If he wants the length to be exactly 10 feet longer than the width, describe the possible dimensions of the pen. Write an inequality and solve. 4b. Write an inequality for the following situation: “This medicine should be stored between 65°F and 75°F for optimum life.”

Algebra 2 Chapter 1 Notes 1.6 Absolute Value Equations and Inequalities

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The universal symbol for “no solution” is

1.6 Solving Absolute Value Equations and Inequalities

A: Solving Absolute Value Equations Perhaps you remember from a previous math class the concept of “absolute value.”

Solve this equation: 5x

Example 1: Solve the following absolute value equation. Be sure to check your answers.

1a. 18 5x 1b. 9 30a

1c. 1

5 103

x 1d. 5 2 4 7 17x

Strategy: x A

Set up 2 equations:

x A or x A


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1e. 5 6 9 0x 1f. 5 3 2 2 7w

1h. 6 5 2 9y 1g. 6 3 2x x

B: Solving Absolute Value Inequalities List the possible integer values of x that make 5x

What do you notice?


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There are two types of absolute value inequalities: “Less than” and “Greater than” “GO L.A.!”can help you remember the difference

Greater than = Rewrite and solve like an Or inequality.

Less than = Rewrite and solve like an And inequality. Example 2: Solve each inequality. Graph the solution.

2a. 4 8 20x 2b. 3 12 6x

2c. 4 1 27s 2d. 10 2 2k


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2e. 3 5 2b Example 3:

Example 4: A chemist is preparing a solution that must contain 3 grams of acid. The acid tolerance

for the solution is described by the inequality 3 0 005.a . What is the weakest acid solution allowed?

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1.4 Solving Equations Expectations · 1.6 Solving Absolute Value Equations and Inequalities A: Solving Absolute Value Equations Perhaps you remember from a previous math class the