Download Solving Equations and Inequalities in One Variable

SOLVING
INEQUALITIES IN ONE
VARIABLE
September 22, 2016
Essential Questions
◦How do I solve an inequality in one
variable?
◦How do I justify the solution of an
inequality?
Outline
◦Adding inequalities
◦Subtracting inequalities
◦Examples
◦Multiplying inequalities
◦Dividing inequalities
◦Examples
Adding Inequalities
Step 1: Determine the operation of the
inequality.
Step 1: x +20 < 39
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 2: x +20 - 20 < 39 - 20
Step 3: Simplify by combining like terms.
Step 3: x +(20 -20) < (39 -20)
x + (0) < (19)
x < 19 (All numbers less than 19)
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
Step 4: x + 20 < 39
(18) + 20 < 39
38 < 39
Subtracting Inequalities
Step 1: Determine the operation of the
inequality.
Step 1: x -12 > 25
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 2: x -12 +12 > 25 +12
Step 3: Simplify by combining like terms.
Step 3: x +(-12 +12) > (25 +12)
x + (0) > (37)
x > 37 (All numbers greater than 37)
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
Step 4: x -12 > 25
(38) - 12 > 25
26 > 25
Example
Step 1: Determine the operation of the
inequality.
Step 1: w +19 < 44
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 2: w +19 - 19 < 44 -19
Step 3: Simplify by combining like terms.
Step 3: w +(19 -19) < (44 -19)
w + (0) < (25)
w< 25 (All numbers less than 25)
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
Step 4: w + 19 < 44
(24) + 19 < 44
43 < 44
Example
Step 1: Determine the operation of the
inequality.
Step 1: n-21 > 16
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 2: n -21 + 21 > 16 + 21
Step 3: Simplify by combining like terms.
Step 3: n +(-21 + 21) > (16 +21)
n + (0) > (37)
n > 37 (All numbers greater than 37)
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
Step 4: n - 21 > 16
(38) - 21 > 16
17 > 16
Example
Step 1: Determine the operation of the
inequality.
Step 1: d - 8 ≥ 24
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 2: d -8 + 8 ≥ 24 + 8
Step 3: Simplify by combining like terms.
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
Step 3: d +(-8 + 8) ≥ (24 +8)
d + (0) ≥ (32)
d ≥ 32 (All numbers greater than or
equal to 32)
Step 4: d - 8 ≥ 24
(33) - 8 ≥ 24
25 ≥ 24
Step 4: d - 8 ≥ 24
(32) - 8 ≥ 24
24 ≥ 24
Multiplying Inequalities
Step 1: Determine the operation of the
inequality.
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 3: Simplify by combining like terms.
Step 1: 17y < 68
Step 2:
17𝑦
( )
17
Step 3:
17𝑦
( )
17
<
68
( )
17
68
( )
17
<
y(1) < (4)
y < 4 (All numbers less than 4)
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
Step 4: 17y < 68
17(3) < 68
51 < 68
Multiplying Inequalities
Step 1: Determine the operation of the
inequality.
Step 1: -6y < 96
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 2: (
Step 3: Simplify by combining like terms.
Step 3: ( ) < ( )
−6
−6
y(1) > (-16) (flip sign when dividing by a
−6𝑦
)
−6
96
−6
<( )
−6𝑦
96
negative number)
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
y > -16 (All numbers greater than -16)
Step 4: -6y < 96
-6(-15) < 96
90 < 96
Dividing Inequalities
Step 1: Determine the operation of the
inequality.
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 1:
< 10
5𝑑
6
5𝑑
6( )
6
Step 2: 6( ) < 6(10)
Step 3:
Step 3: Simplify by combining like terms.
Step 4:
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
5𝑑
6
< 6(10)
5d < (60)
5𝑑
60
<
5
5
d<12 (All numbers less than 12)
5𝑑
< 10
6
5(6)
< 10
6
30
< 10
6
5 < 10
Dividing Inequalities
Step 1: Determine the operation of the
inequality.
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 1:
Step 2:
Step 3:
4𝑚
−
>8
3
4𝑚
-3(- ) >
3
4𝑚
-3(- ) >
3
-3(8) (flip sign when
multiplying by a negative number)
4m < (-24)
4𝑚
−24
<
4
4
m<-6 (All numbers less than -6)
Step 3: Simplify by combining like terms.
4𝑚
>8
3
4(−9)
>8
3
−36
>8
3
Step 4: −
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
-3(8)
-
12 > 8
Example
Step 1: Determine the operation of the
inequality.
Step 1: 12 ≥ 4a
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 2: ( ) ≥ ( )
Step 3: Simplify by combining like terms.
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
Step 3:
12
4
4𝑎
4
12
( )
4
4𝑎
( )
4
≥
3≥a
a ≤ 3 (All numbers less than or
equal to 3)
Step 4: 12 ≥ 4a
12 ≥ 4(2)
12 ≥ 8
Example
Step 1: Determine the operation of the
inequality.
Step 1: -5y < 60
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 2: (
Step 3: Simplify by combining like terms.
Step 3: ( ) < ( )
−5
−5
y(1) > (-12) (flip sign when dividing by a
−5𝑦
)
−5
60
−5
<( )
−5𝑦
60
negative number)
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
y > -12 (All numbers greater than -12)
Step 4: -5y < 60
-5(-11) < 60
55 < 60
Example
Step 1: Determine the operation of the
inequality.
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
Step 3: Simplify by combining like terms.
Step 1: 7 <
7𝑑
6
Step 2: (6)7 < 6( )
7𝑑
Step 3: (6)7< 6( )
6
42 < 7d
42
7𝑑
<
7
7
6<d
d > 6 (All numbers greater than 6)
Step 4: 7 <
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
7𝑑
6
7<
7𝑑
6
7 (12)
6
84
6
7<
7 < 14
Example
Step 1: Determine the operation of the
inequality.
Step 2: Isolate variable by using inverse
operation. Use the same operation on
both sides of inequality.
2𝑚
>6
5
2𝑚
-5(- ) >
5
2𝑚
-5(- ) >
5
Step 1: −
Step 2:
Step 3:
-5(6) (flip sign when
multiplying by a negative number)
2m < (-30)
2𝑚
−30
<
2
2
m<-15 (All numbers less than -15)
Step 3: Simplify by combining like terms.
2𝑚
>6
5
2(−20)
>6
5
−40
>6
5
Step 4: −
Step 4: Verify solution by substituting an
answer that is appropriate for the
variable.
-5(6)
-
8>6