Objective - To solve equations over given replacement sets. Equalities Inequalities = Equals- is the...
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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
~
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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.
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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
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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
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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:
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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”,
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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
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=x + 3 7- 3 Heavier
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=x 7Heavier
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=x7 Heavier
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=x
7 Heavier
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=x
7 Heavier
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=x + 3 7- 3 - 3
=x 4
Algebraically,
x + 3 = 7-3 -3x = 4
x + 3 = 7
x = 4x + 3 - 3 = 7 - 3
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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.
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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
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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
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Multiplication Property of Equality
If a = b, then a c = b c.
orxIf n,m
xthen nm
or x mn.
m m
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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
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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
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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
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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.
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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.