name the property: if a > b , then a + c > b + c
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Exercise. Name the property: If a > b , then a + c > b + c. Addition Property of Inequality. Exercise. If a > b and c > 0, then ac ___ bc. >. Exercise. If a > b and c < 0, then ac ___ bc.TRANSCRIPT
Name the property: If a > b, then a + c > b + c.Name the property: If a > b, then a + c > b + c.
Addition Property of Inequality
Addition Property of Inequality
ExerciseExercise
If a > b and c > 0, then ac ___ bc.If a > b and c > 0, then ac ___ bc.
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ExerciseExercise
If a > b and c < 0, then ac ___ bc.If a > b and c < 0, then ac ___ bc.
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ExerciseExercise
Solve x + 3 < 5.Solve x + 3 < 5.
x < 2x < 2
ExerciseExercise
Solve β 2x > 8.Solve β 2x > 8.
x < β 4x < β 4
ExerciseExercise
Properties of InequalitiesProperties of Inequalities
PropertyProperty
ExampleExample
If a < b, then a + c < b + c.
6 < 7 and 6 + 3 < 7 + 3; i.e., 9 < 10
Properties of InequalitiesProperties of Inequalities
PropertyProperty
ExampleExample
If a < b, then a β c < b β c.
β 4 < β 2 and β 4 β 5 < β 2 β 5; i.e., β 9 < β 7
Properties of InequalitiesProperties of Inequalities
PropertyProperty
ExampleExample
If a < b and c > 0, then ac < bc.
2 < 5 and 2(3) < 5(3); i.e., 6 < 15
Properties of InequalitiesProperties of Inequalities
PropertyProperty
ExampleExample
If a < b and c < 0, then ac > bc.
2 < 5 and 2(β 3) < 5(β 3); i.e., β 6 > β 15
Properties of InequalitiesProperties of Inequalities
PropertyProperty
ExampleExample
If a < b and c > 0, then < .If a < b and c > 0, then < .a
cac
bcbc
4 < 8 and < ; i.e., 2 < 4
4 < 8 and < ; i.e., 2 < 4
4242
8282
Properties of InequalitiesProperties of Inequalities
PropertyProperty
ExampleExample
If a < b and c < 0, then > .If a < b and c < 0, then > .a
cac
bcbc
4 < 8 and ; i.e., β 2 > β 4
4 < 8 and ; i.e., β 2 > β 4
4β 24
β 28
β 28
β 2>>
Reverse signs if:Reverse signs if:
β’ multiplying by a negativeβ’ multiplying by a negativex4x4
β > 10 x < β 40β > 10 x < β 40
β’ dividing by a negativeβ’ dividing by a negative
β 3x < 6 x > β 2β 3x < 6 x > β 2
Solve β 4x + 3 > 23, and graph the solution.Solve β 4x + 3 > 23, and graph the solution.
x < β 5x < β 5
β 4x + 3 β 3 > 23 β 3β 4x + 3 β 3 > 23 β 3β 4x > 20β 4β 4 β 4β 4
00β 1β 1β 2β 2β 3β 3β 4β 4β 5β 5β 6β 6
Example 1Example 1
Solve β x + 8 β€ 23, and graph the solution.Solve β x + 8 β€ 23, and graph the solution.
x β₯ β 20x β₯ β 20
3434
β x + 8 β 8 β€ 23 β 8β x + 8 β 8 β€ 23 β 83434
β x β€ 15β x β€ 153434
4343
ββ 4343
ββ55
Example 2Example 2
101000β 10β 10β 20β 20β 30β 30
Solve β x + 8 β€ 23, and graph the solution.Solve β x + 8 β€ 23, and graph the solution.
3434
x β₯ β 20x β₯ β 20
Example 2Example 2
β 2r > 27β 2r > 27
Solve β 2(r + 4) > 19, and graph the solution.Solve β 2(r + 4) > 19, and graph the solution.
r < β 13.5r < β 13.5
β 2r β 8 > 19β 2r β 8 > 19
β 2β 2β 2β 2
β 2r β 8 + 8 > 19 + 8β 2r β 8 + 8 > 19 + 8
Example 3Example 3
r < β 13.5r < β 13.5
β 12β 12β 13β 13β 14β 14β 15β 15β 16β 16
Solve β 2(r + 4) > 19, and graph the solution.Solve β 2(r + 4) > 19, and graph the solution.
Example 3Example 3
Solve 2x β₯ β 2x + 8.Solve 2x β₯ β 2x + 8.
x β₯ 2x β₯ 244 44
2x + 2x β₯ β 2x + 2x + 82x + 2x β₯ β 2x + 2x + 84x β₯ 84x β₯ 8
Example 4Example 4
Solve 3x β 1 < x + 9.Solve 3x β 1 < x + 9.
x < 5x < 522 22
3x β 1 β x < x + 9 β x3x β 1 β x < x + 9 β x2x β 1 < 92x β 1 < 9
2x β 1 + 1 < 9 + 12x β 1 + 1 < 9 + 12x < 102x < 10
Example 5Example 5
Solve 5x β 8 β€ 9x + 2.Solve 5x β 8 β€ 9x + 2.
x β₯ β 2.5x β₯ β 2.5β 4β 4 β 4β 4
5x β 8 β 9x β€ 9x + 2 β 9x5x β 8 β 9x β€ 9x + 2 β 9xβ 4x β 8 β€ 2β 4x β 8 β€ 2
β 4x β 8 + 8 β€ 2 + 8β 4x β 8 + 8 β€ 2 + 8β 4x β€ 10β 4x β€ 10
Example 6Example 6
Solve 5x + 3 < β 7.Solve 5x + 3 < β 7.
x < β 2x < β 2
ExampleExample
Solve β 3x + 5 > 17.Solve β 3x + 5 > 17.
x < β 4x < β 4
ExampleExample
Solve 2x β 12 < 7x + 13.Solve 2x β 12 < 7x + 13.
x > β 5x > β 5
ExampleExample
Solve 2(n + 7) > β 3n + 12.Solve 2(n + 7) > β 3n + 12.
n > β n > β 2525
ExampleExample
Solve 3(y β 12) < β 2(y β 9) + 1.Solve 3(y β 12) < β 2(y β 9) + 1.
y < 11y < 11
ExampleExample
Solve 2.5(x β 3) β 3(x β 2.5) > 2x.Solve 2.5(x β 3) β 3(x β 2.5) > 2x.
x < 0x < 0
ExampleExample
Solve 8(0.75x β 0.375) < 12(1.25x + 0.5).Solve 8(0.75x β 0.375) < 12(1.25x + 0.5).
x > β 1x > β 1
ExampleExample
Solve + 3 β₯ 12.Solve + 3 β₯ 12.
b β₯ 45b β₯ 45
b5b5
ExampleExample
Solve β₯ 12.Solve β₯ 12.
b β₯ 57b β₯ 57
b + 35
b + 35
ExampleExample
Solve β€ .Solve β€ .
a β₯ 39a β₯ 39
a + 37
a + 37
a β 95
a β 95
ExampleExample
Using the inequality ax β b β€ c, and assuming that a, b, and c are real numbers with a β 0, solve the inequality for x. Be careful to account for all possible values of a, b, and c.
Using the inequality ax β b β€ c, and assuming that a, b, and c are real numbers with a β 0, solve the inequality for x. Be careful to account for all possible values of a, b, and c.
ExerciseExercise
Using the inequality + s > t,
and assuming that r, s, and t are real numbers, solve the inequality for x. Be careful to account for all possible values of r, s, and t.
Using the inequality + s > t,
and assuming that r, s, and t are real numbers, solve the inequality for x. Be careful to account for all possible values of r, s, and t.
xrxr
ExerciseExercise