Objective - To solve equations over
given replacement sets.
Equalities
= Equals- is the
same as
 Congruent- same
size and shape
~ Similar- same
shape
Inequalities
< Is less than
> Is greater than
 Is less than or
equal to
 Approx. equal to
= Not equal to
Expressions vs. Equations
Sentences
Expressions
Numerical
Variable
2+3
5(8) - 4
x+7
8 - 3y
Equations
2+3=5
4 + 2(3) = 10
x - 4 = 13
11= 3 + 2m
Inequalities
9-5>3
6y - 4 < 8
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
Sentence, open sentence, equation
2) 7 < 2(5) + 3
Sentence, inequality
3) 5x - 2
Expression
4) 6m + 2 > 3
Sentence, open sentence, inequality
State whether each sentence is true, false ,or open.
1) 8 + 5 = 13
True
5) 14 - 2(3) = 8
True
2) 2x - 1 = 9
6) 9 = 7 + 4y
Open
Open
3) 17 = 3(5) + 1
False
4) 3 = 7(2) - 5
False
7) 13 - 2 = 9
False
8) t + t = 5(2) + 1
Open
Replacement Set
Equation
0,1, 2,3
x x 0
Try each key:
Solution Set
0 0  0
2
1 1  0
2
2 2  0
2
3 3  0
2
2
 0 ,1
Solve the given equation using the replacement
set {0, 1, 2, 3, 4}.
1) 6 - x = 2
{4}
2) 2x + 1 = 5
{2}
3) 5x = 15
{3}
4) 11 = 4x + 3
{2}
5) 2x = x + x
{0, 1, 2, 3, 4}
6) 9 = 7 + 2y
{1}
7) x + 5 = 27
{ },  , or “No solution”
8) x + 2 = x
{ },  , or “No solution”
Equivalent Equations
Addition Property of Equality
If a = b, then a + c = b + c
or
Given a = b
and c = c
then a + c = b + c
Subtraction Property of Equality
If a = b, then a - c = b - c
or
Given a = b
and c = c
then a - c = b - c
x+3
-3
=
7
Heavier
x
=
7
Heavier
x
=
7
Heavier
x
=
7
Heavier
x
=
7
Heavier
x+3
-3
=
7
-3
x
=
4
Algebraically,
x+3=7
-3 -3
x=4
x+3=7
x+3-3=7-3
x=4
1) Goal: Isolate the variable on one side
of the equation.
2) Always perform the same operation to
both 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 set
is the set of whole numbers.
1) x + 3 = 10
-3 -3
x=7
4) 13 = x + 5
-5
-5
8=x
2) y - 8 = 11
+8 +8
y = 19
5) 12 = n - 3
+3
+3
15 = n
3) n + 5 = 11
-5 -5
n=6
6) 11 + 3 = k
14 = k
Translate the sentence into an equation and solve.
1) The sum of k and 13 is 28.
k + 13 = 28
- 13 - 13
k = 15
2) Five is the difference of t and 4.
5=t-4
+4
+4
9=t
Multiplication Property of Equality
If a = b, then a c = b c.
or
x
x
If
 n, then m   n  m
m
m
or x  mn.
Solve given the replacement set is the set of whole
numbers.
1) x  4
3
x
3  4  3
3
x  12
y
2)
 16
2
y
2   16  2
2
y  32
3) m  10
5
m
5   10  5
5
m  50
k 7
4
k
4  74
4
k  28
4)
Division Property of Equality
If a  b, then a  c  b  c.
or
If x  m  n, then x  m  n
m
m
n
or x  .
m
Solve given the replacement set is the set of whole
numbers.
1) 5x  20
3) 24  8t
5x  20
5
5
x4
2) 36  3y
36  3y
3
3
12  y
24  8t
8
8
3 t
4) 4k  18
4k  18
4
4
1
9
18
 or 4
k
2
2
4
No solution
Each pair of equations is equivalent. Tell what
was done to the first equation to get the second.
1) 2x  4  20
2x  16
Four was subtracted from both sides.
2) 3y  9  13
3y  22
Nine was added to both sides.
x
3)  11
4
x  44
Each side was multiplied by 4.
Each pair of equations is equivalent. Tell what
was done to the first equation to get the second.
4) 24  6x
4x
Each side was divided by 6.
5) 6  7m  48
7m  42
Six was subtracted from both sides.
m
6) 12 
7
84  m
Each side was multiplied by 7.