1.6 Solving Inequalities.. Trichotomy Property a b.
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Transcript of 1.6 Solving Inequalities.. Trichotomy Property a b.
1.6Solving Inequalities.
Trichotomy Property
a < b a = b a > b
Nonnegative Property
Transitive Property of Inequalities
If a < b and b < c,then a < c.
If a > b and b > c, then a > c.
Addition Property of Inequalities
If a < b, then a + c < b + c.
If a > b, then a + c > b + c.
Multiplication Property of Inequalities
If a < b and if c > 0, then ac < bc.
If a < b and if c < 0, then ac > bc.
If a > b and if c > 0, then ac > bc.
If a > b and if c < 0, then ac < bc.
3 < 5
2(3) < 2(5)
3 < 5
-2(3) > -2(5)
-6 > -10
Reciprocal Property for Inequalities
A closed interval denoted by [a, b], consists of all real numbers x for which a < x < b.
[ ]a b
An open interval, denoted (a, b), consists of all real numbers x for which a < x < b.
( ) a b
A half-open, or half-closed interval is (a, b], consisting of all real numbers x for which a < x < b.
( ]a b
A half-open, or half-closed interval is [a, b), consisting of all real numbers x for which a < x < b.
[ )a b
[a
(a
[ )−3 2,
[ )-3 20
Write the inequality -3 < x < 2 using interval notation. Illustrate the inequality using a real number line.
Steps for Solving Inequalities Graphically
• Write the inequality in one of the following forms:
• Graph Y1 and Y2 on the same screen.
• If inequality is of the form Y1 < Y2, determine on what interval Y1 is below Y2.
• Similarly for Y1 > Y2.
• If inequality is not strict include the endpoints of the intervals.
Y1<Y2, Y1>Y2, Y1≤Y2, Y1≥Y2
Inequalities Involving Absolute Value
3 1 5x − <
− < − <5 3 1 5x
− < <4 3 6x
− < <43
2x
Solution set: (-4/3, 2)
-2 -1 0 1 2 3 ( )
Solve absolute value inequality
Inequalities Involving Absolute Value
Solve Inequality