Download Scholarship Algebra II 1-4: Solving Inequalities Day 1 Properties of

Scholarship Algebra II
1-4: Solving Inequalities Day 1
Properties of Inequalities
Let a, b, and c represent real numbers.
Transitive Property
If a " b and b " c , then a " c .
Addition Property
If a " b , then a + c " b + c .
Subtraction Property
If a " b , then a " c # b " c .
Multiplication Property
If a " b and c > 0 , then ac " bc .
!
!
!
If a " b and c < 0 , then ac " bc .
!
!
a b
Division Property
If a!" b and c > 0 , then " .
!
c c
!
!
!
a b
!
< 0 , then " .
a " b and c !
If !
c c
!
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Ex. 1:
!
!
Solve and graph the inequality.
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14 " 4 y # 38
-7
-6
-5
-4
-3
-2
-1
0
1
2
Ex. 2:
Solve and graph the inequality.
2(m " 3) + 7 < 21
2(m " 3) < 21" 7
3
14
m"3<
2
m < 7+ 3
m < 10
4
5
6
7
8
9
10
11
12
"4 y # 38 "14
24
y$
"4
y $ "6
!
!
Ex. 3:
Solve and graph the inequality.
9( x + 2) > 9( x " 3)
x+2> x"3
2 > "3
All real numbers are solutions.
!
Ex. 4:
Solve and graph the inequality.
"6(2x "10) + 12x # 180
"12x + 60 + 12x # 180
60 # 180
No real solutions.
!
Ex. 5:
Solve and graph the inequality.
2
( x "12) < x + 8
3
2(x "12) < 3(x + 8)
2x " 24 < 3x + 24
"48 < x
!
-54
-52
-50
-48
-46
Ex. 6:
The length of a picture frame is 3 in. greater than the width. The perimeter is less than 52 in.
Describe the possible dimensions.
x
2x + 2( x + 3) < 52
x+3
!
2x + 2x + 6 < 52
4 x < 46
x < 11.5
The width is less than 11.5 in. and greater than 0 in.
The length is 3 in. longer than the width, so it is between
3 in. and 14.5 in.
Ex. 7:
Find the lesser of two consecutive integers whose sum is greater than 16.
x + (x + 1) > 16
2x + 1 > 16
2x > 15
x > 7.5
!
The smallest pair of consecutive integers that meets the condition is 8, 9. There are many other
possibilities (any integer equal to or greater than 8).