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
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Solve – 2x > 8.Solve – 2x > 8.
x < – 4x < – 4
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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 – c < b – c.
– 4 < – 2 and – 4 – 5 < – 2 – 5; i.e., – 9 < – 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(– 3) < 5(– 3); i.e., – 6 > – 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., – 2 > – 4
4 < 8 and ; i.e., – 2 > – 4
4– 24
– 28
– 28
– 2>>
Reverse signs if:Reverse signs if:
• multiplying by a negative• multiplying by a negativex4x4
– > 10 x < – 40– > 10 x < – 40
• dividing by a negative• dividing by a negative
– 3x < 6 x > – 2– 3x < 6 x > – 2
Solve – 4x + 3 > 23, and graph the solution.Solve – 4x + 3 > 23, and graph the solution.
x < – 5x < – 5
– 4x + 3 – 3 > 23 – 3– 4x + 3 – 3 > 23 – 3– 4x > 20– 4– 4 – 4– 4
00– 1– 1– 2– 2– 3– 3– 4– 4– 5– 5– 6– 6
Example 1Example 1
Solve – x + 8 ≤ 23, and graph the solution.Solve – x + 8 ≤ 23, and graph the solution.
x ≥ – 20x ≥ – 20
3434
– x + 8 – 8 ≤ 23 – 8– x + 8 – 8 ≤ 23 – 83434
– x ≤ 15– x ≤ 153434
4343
–– 4343
––55
Example 2Example 2
101000– 10– 10– 20– 20– 30– 30
Solve – x + 8 ≤ 23, and graph the solution.Solve – x + 8 ≤ 23, and graph the solution.
3434
x ≥ – 20x ≥ – 20
Example 2Example 2
– 2r > 27– 2r > 27
Solve – 2(r + 4) > 19, and graph the solution.Solve – 2(r + 4) > 19, and graph the solution.
r < – 13.5r < – 13.5
– 2r – 8 > 19– 2r – 8 > 19
– 2– 2– 2– 2
– 2r – 8 + 8 > 19 + 8– 2r – 8 + 8 > 19 + 8
Example 3Example 3
r < – 13.5r < – 13.5
– 12– 12– 13– 13– 14– 14– 15– 15– 16– 16
Solve – 2(r + 4) > 19, and graph the solution.Solve – 2(r + 4) > 19, and graph the solution.
Example 3Example 3
Solve 2x ≥ – 2x + 8.Solve 2x ≥ – 2x + 8.
x ≥ 2x ≥ 244 44
2x + 2x ≥ – 2x + 2x + 82x + 2x ≥ – 2x + 2x + 84x ≥ 84x ≥ 8
Example 4Example 4
Solve 3x – 1 < x + 9.Solve 3x – 1 < x + 9.
x < 5x < 522 22
3x – 1 – x < x + 9 – x3x – 1 – x < x + 9 – x2x – 1 < 92x – 1 < 9
2x – 1 + 1 < 9 + 12x – 1 + 1 < 9 + 12x < 102x < 10
Example 5Example 5
Solve 5x – 8 ≤ 9x + 2.Solve 5x – 8 ≤ 9x + 2.
x ≥ – 2.5x ≥ – 2.5– 4– 4 – 4– 4
5x – 8 – 9x ≤ 9x + 2 – 9x5x – 8 – 9x ≤ 9x + 2 – 9x– 4x – 8 ≤ 2– 4x – 8 ≤ 2
– 4x – 8 + 8 ≤ 2 + 8– 4x – 8 + 8 ≤ 2 + 8– 4x ≤ 10– 4x ≤ 10
Example 6Example 6
Solve 5x + 3 < – 7.Solve 5x + 3 < – 7.
x < – 2x < – 2
ExampleExample
Solve – 3x + 5 > 17.Solve – 3x + 5 > 17.
x < – 4x < – 4
ExampleExample
Solve 2x – 12 < 7x + 13.Solve 2x – 12 < 7x + 13.
x > – 5x > – 5
ExampleExample
Solve 2(n + 7) > – 3n + 12.Solve 2(n + 7) > – 3n + 12.
n > – n > – 2525
ExampleExample
Solve 3(y – 12) < – 2(y – 9) + 1.Solve 3(y – 12) < – 2(y – 9) + 1.
y < 11y < 11
ExampleExample
Solve 2.5(x – 3) – 3(x – 2.5) > 2x.Solve 2.5(x – 3) – 3(x – 2.5) > 2x.
x < 0x < 0
ExampleExample
Solve 8(0.75x – 0.375) < 12(1.25x + 0.5).Solve 8(0.75x – 0.375) < 12(1.25x + 0.5).
x > – 1x > – 1
ExampleExample
Solve + 3 ≥ 12.Solve + 3 ≥ 12.
b ≥ 45b ≥ 45
b5b5
ExampleExample
Solve ≥ 12.Solve ≥ 12.
b ≥ 57b ≥ 57
b + 35
b + 35
ExampleExample
Solve ≤ .Solve ≤ .
a ≥ 39a ≥ 39
a + 37
a + 37
a – 95
a – 95
ExampleExample
Using the inequality ax – b ≤ c, and assuming that a, b, and c are real numbers with a ≠ 0, solve the inequality for x. Be careful to account for all possible values of a, b, and c.
Using the inequality ax – b ≤ c, and assuming that a, b, and c are real numbers with a ≠ 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
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