Objective - To solve equations over given replacement sets. Equalities Inequalities = Equals- is the...

Objective - To solve equations over given replacement sets.

Equalities Inequalities

= Equals- is the same as

Congruent- same size and shape

Similar- same shape

< Is less than

> Is greater than

Is less than or equal to

Approx. equal to

= Not equal to

~

Expressions vs. Equations

Numerical

Variable

Expressions Equations Inequalities

2 + 35(8) - 4

x + 78 - 3y

2 + 3 = 54 + 2(3) = 10

x - 4 = 1311= 3 + 2m

9 - 5 > 3

6y - 4 < 8

Sentences

Open sentences

Open sentences have solutions and can be solved.

Identify each as an expression, sentence, open sentence, equation, or inequality.

1) 3x + 5 = 11

2) 7 < 2(5) + 3

3) 5x - 2

4) 6m + 2 > 3

Sentence, open sentence, equation

Sentence, inequality

Expression

Sentence, open sentence, inequality

State whether each sentence is true, false ,or open.

1) 8 + 5 = 13

2) 2x - 1 = 9

3) 17 = 3(5) + 1

4) 3 = 7(2) - 5

5) 14 - 2(3) = 8

6) 9 = 7 + 4y

7) 13 - 2 = 9

8) t + t = 5(2) + 1

True

Open

False

False

True

Open

False

Open

Replacement Set

0,1,2,3

Equation2x x 0

Solution Set

020 0 0 21 1 0

,1

22 2 0 23 3 0

Try each key:

Solve the given equation using the replacementset {0, 1, 2, 3, 4}.

1) 6 - x = 2

2) 2x + 1 = 5

3) 5x = 15

4) 11 = 4x + 3

5) 2x = x + x

6) 9 = 7 + 2y

7) x + 5 = 27

8) x + 2 = x

{4}

{2}

{3}

{2}

{0, 1, 2, 3, 4}

{1}

{ } , or “No solution”,

{ } , or “No solution”,

Addition Property of EqualityIf a = b, then a + c = b + c

orGiven a = b

and c = cthen a + c = b + c

Subtraction Property of EqualityIf a = b, then a - c = b - c

orGiven a = b

and c = cthen a - c = b - c

Equivalent Equations

=x + 3 7- 3 Heavier

=x 7Heavier

=x7 Heavier

=x

7 Heavier

=x

7 Heavier

=x + 3 7- 3 - 3

=x 4

Algebraically,

x + 3 = 7-3 -3x = 4

x + 3 = 7

x = 4x + 3 - 3 = 7 - 3

1) Goal: Isolate the variable on one side of the equation.

2) Always perform the same operation toboth sides of an equation.

3) To undo an operation, perform its opposite operation to both sides of the equation.

Solve the equations below. The replacement setis the set of whole numbers.

1) x + 3 = 10- 3 - 3

x = 7

2) y - 8 = 11+8 +8

y = 19

3) n + 5 = 11- 5 - 5

n = 6

4) 13 = x + 5- 5 - 5

8 = x

5) 12 = n - 3+3 +315 = n

6) 11 + 3 = k

14 = k

Translate the sentence into an equation and solve.

1) The sum of k and 13 is 28.

2) Five is the difference of t and 4.

k + 13 = 28- 13 - 13

k = 15

5 = t - 4+4 +4

9 = t

Multiplication Property of Equality

If a = b, then a c = b c.

orxIf n,m

xthen nm

or x mn.

m m

Solve given the replacement set is the set of wholenumbers.

1)

2)

3)

4)

x 43

x3 4 33

x 12

y16

2

y2 16 2

2

y 32

m 105

m5 10 55

m 50

k 74

k4 7 44

k 28

Division Property of Equality

or

If x m n, then

nor x .m

If a b, then a c b c.

x m n m m

1)

2)

3)

4)

5x 205x 20

x 45 5

36 3y36 3y

12 y3 3

24 8t24 8t

3 t8 8

4k 184k 18

18k4

4 4

92

1or 42

Solve given the replacement set is the set of wholenumbers.

No solution

Each pair of equations is equivalent. Tell whatwas done to the first equation to get the second.

1) 2x 4 20 2x 16

2) 3y 9 13 3y 22

x3) 114

x 44

Four was subtracted from both sides.

Nine was added to both sides.

Each side was multiplied by 4.

Each pair of equations is equivalent. Tell whatwas done to the first equation to get the second.

4) 24 6x 4 x

5) 6 7m 48 7m 42

m6) 127

84 m

Each side was divided by 6.

Six was subtracted from both sides.

Each side was multiplied by 7.