Algebra IIAlgebra II

1.3 Solving Equations1.3 Solving Equations

In math, what are variables? In math, what are variables?

Letters that represent numbers.Letters that represent numbers.

In math, what is an equation?In math, what is an equation?

A mathematical expression to include an A mathematical expression to include an equal sign.equal sign.

Open sentenceOpen sentence – a mathematical sentence – a mathematical sentence containing one or more variables.containing one or more variables.

Write an algebraic expression to represent each Write an algebraic expression to represent each verbal expression.verbal expression.

1A) 7 less than a number.1A) 7 less than a number. x – 7x – 7 B) three times the square of a number.B) three times the square of a number. 3x3x²² C) the cube of a number increased by 4 times the C) the cube of a number increased by 4 times the

same number.same number. x³ + 4xx³ + 4x D) twice the sum of a number and 5.D) twice the sum of a number and 5. 2 (x + 5)2 (x + 5)

2A) 10 = 12 – 22A) 10 = 12 – 2 Ten is equal to twelve minus two.Ten is equal to twelve minus two. B) n + (-8) = -9B) n + (-8) = -9 The sum of a number and -8 is -9.The sum of a number and -8 is -9. C) n/6 = nC) n/6 = n² ² A number divided by six is equal to thatA number divided by six is equal to that number squared.number squared.

Properties of EqualityProperties of Equality

PropertyProperty SymbolsSymbols ExamplesExamples

ReflexiveReflexive A = AA = A -7 + n = -7 + n-7 + n = -7 + n

SymmetricSymmetric If a= b, then b = aIf a= b, then b = a If 3 = 5x – 6, If 3 = 5x – 6, then 5x – 6 = 3then 5x – 6 = 3

TransitiveTransitive If a = b, and b = c, If a = b, and b = c, then a = cthen a = c

If 2x+1 = 7 and If 2x+1 = 7 and 7 = 5x -8, then 7 = 5x -8, then 2x+1 = 5x-82x+1 = 5x-8

SubstitutionSubstitution If a = b, then a If a = b, then a can replace bcan replace b

If (4+5)m = 18, If (4+5)m = 18, then 9m = 18then 9m = 18

Name the property illustrated.Name the property illustrated. 3. 3m = 5n, and 5n = 10p, then 3m = 10p3. 3m = 5n, and 5n = 10p, then 3m = 10p Transitive propertyTransitive property 4. If -11a + 2 = -3a, then -3a= -11a + 24. If -11a + 2 = -3a, then -3a= -11a + 2 Symmetric propertySymmetric property

Warmup 8/31/12Warmup 8/31/12

1. Write an example of the symmetric 1. Write an example of the symmetric property.property.

2. Write an expression for 2(2x -6).2. Write an expression for 2(2x -6).

ProblemsProblems

Solve these equations.Solve these equations. 1. X + 14 = -71. X + 14 = -7

-14 -14-14 -14

X = -21X = -21

2. 7 + 3x = 492. 7 + 3x = 49

-7 -7-7 -7

3x = 423x = 42

x = 14x = 14

3. 1.8a – 5 = -2.33. 1.8a – 5 = -2.3 +5 +5 +5 +5 1.8a = 2.71.8a = 2.7 a = 1.5a = 1.5

4. -4(k + 7) = -124. -4(k + 7) = -12 -4k - 28 = -12-4k - 28 = -12 +28 +28+28 +28 -4k = 16-4k = 16 k = -4k = -4

5.5. 7x + x – 3x = -247x + x – 3x = -24 5x = -245x = -24 x = -4.8x = -4.8

6. 3d + 7 = 6d + 56. 3d + 7 = 6d + 5 -6d -6d-6d -6d -3d + 7 = 5-3d + 7 = 5 -7 -7-7 -7 -3d = -2-3d = -2 d = 2/3d = 2/3

InequalitiesInequalities

Rules for Multiplication / DivisionRules for Multiplication / Division– When you multiply / divide both sides by a When you multiply / divide both sides by a

negative, you must __________ the sign.negative, you must __________ the sign. Try these and Graph on a number line:Try these and Graph on a number line: 1. a + 2.5 < 3.7 1. a + 2.5 < 3.7

2. 5 2. 5 ≥ 3x≥ 3x

Solving for a variableSolving for a variable

Place a box around the number you are Place a box around the number you are solving for. Move everything to the other solving for. Move everything to the other side using equation rules.side using equation rules.

Example1: Solve for r when d = rtExample1: Solve for r when d = rt

Example 2: Solve for h when V = Example 2: Solve for h when V = ⅓iir²h⅓iir²h

InequalitiesInequalities

Rules for Multiplication / DivisionRules for Multiplication / Division– When you multiply / divide both sides by a negative, you must When you multiply / divide both sides by a negative, you must flipflip

the sign.the sign.

Try these and graph on a number line: Try these and graph on a number line: 1. 1. a + 2.5 < 3.7a + 2.5 < 3.7 - 2.5 -2.5- 2.5 -2.5 a < 1.2a < 1.2

2. 2. 5 5 ≥ 3x≥ 3x 3x ≤ 53x ≤ 5 x ≤ 5/3x ≤ 5/3

Solve and GraphSolve and Graph

3. 14 – 8n 3. 14 – 8n ≤ 0≤ 0 -14 -14-14 -14 -8n ≤ -14-8n ≤ -14 n ≥ 7/4n ≥ 7/44.4. 2 (4t + 9) < 182 (4t + 9) < 18 8t + 18 < 188t + 18 < 18 -18 -18-18 -18 8t < 08t < 0 t < 0t < 0

Warmup 9/4/12Warmup 9/4/12

1. Solve 4(x + 2) = 121. Solve 4(x + 2) = 12

2. Solve and graph: 2x + 6 < 122. Solve and graph: 2x + 6 < 12

Word ProblemsWord Problems

7. Austin and Anna go to the West Bladen Bowling 7. Austin and Anna go to the West Bladen Bowling alley. They have a total of $31 to spend on their alley. They have a total of $31 to spend on their bowling adventure. It costs $3.50 to rent shoes bowling adventure. It costs $3.50 to rent shoes and $6.00 per game to bowl. Find the maximum and $6.00 per game to bowl. Find the maximum number of games that they can bowl if they each number of games that they can bowl if they each rent shoes.rent shoes.

What information should the variable represent?What information should the variable represent?The amount per game.The amount per game.

Write the equation for the word problem and solve.Write the equation for the word problem and solve.31 = 6.00x + 7.0031 = 6.00x + 7.00 x = 4 gamesx = 4 games

8. Suppose Brandon and Barbie also go to 8. Suppose Brandon and Barbie also go to West Bladen Bowling alley but they have a West Bladen Bowling alley but they have a total of $60 to spend on bowling. Find the total of $60 to spend on bowling. Find the maximum number of games that they can maximum number of games that they can bowl if they rent shoes. Let n be the bowl if they rent shoes. Let n be the variable this time.variable this time.

Write an equation for this word problem and Write an equation for this word problem and solve.solve. 60 = 6.00x + 7.0060 = 6.00x + 7.00 x = 8 gamesx = 8 games

9. Larry spent $10,734 9. Larry spent $10,734 on his living expenses on his living expenses last year. Most of last year. Most of these expenses are these expenses are listed to the right. If he listed to the right. If he paid rent 12 times last paid rent 12 times last year, how much was year, how much was Larry’s rent each Larry’s rent each month?month?

ExpenseExpense Annual Annual CostCost

ElectricElectric $622$622

GasGas $428$428

WaterWater $240$240

Renter’s Renter’s

InsuranceInsurance

$144$144

AssignmentAssignment

In your workbook:In your workbook: Do page 3 (13-26 all)Do page 3 (13-26 all) Do page 5 (1-16 all)Do page 5 (1-16 all)

***Remember to show ALL work!!!***Remember to show ALL work!!!