a.8 interval notation; solving inequalities€¦ · a.8 interval notation and solving inequalities...
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A.8 Interval Notation and Solving Inequalities 2010
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September 22, 2010
Sep 11:52 PM
A.8 Interval Notation; Solving Inequalities
Objective:• Use interval notation• Use properties of inequalities• Solve Linear Inequalities• Solve Combined Inequalities• Solve Absolute Value Inequalities
A.8 Interval Notation and Solving Inequalities 2010
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Interval NotationLet a and b represent two real numbers with a < b.
Closed Interval: written [a, b], consists of all real numbers x for which a < x < b
Open Interval: written (a, b), consists of all real numbers x for which a < x < b
Half-Open or Half-Closed Intervals: written (a, b], consists of all real numbers x for which a < x < b written [a, b), consists of all real numbers x for which a < x < b
1 02345 1 2 3 4 5
1 02345 1 2 3 4 5
1 02345 1 2 3 4 5
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Interval Notation with Infinity
The symbol ∞ (infinity) indicates unboundedness in a positive direction.The symbol -∞ (negative infinity) indicates unboundedness in a negative direction.
Match the following intervals with the appropriate inequalities.Note that ∞ and -∞ are never included as endpoints since they are not real numbers.
1. [a, ∞)
2. (a, ∞)
3. (-∞, a]
4. (-∞, a)
5. (-∞, ∞)
x > a
x > a
x < a
x < a
x = R
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Examples:Write each inequality using interval notation.
1. 1 < x < 3 2. -4 < x < 0 3. x > 5 4. x < 1
Write each interval as an inequality involving x.
5. [1, 4) 6. (2, ∞) 7. [2, 3] 8. (-∞, -3]
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Properties of InequalitiesNonnegative Property: (for any real number a) a2 > 0
Addition Property of Inequalities: (for real numbers a, b and c)if a < b, then a + c < b + cif a > b, then a + c > b + c
Multiplication Properties for Inequalities: (for real numbers a, b and c)if a < b and if c > 0, then ac < bcif a < b and if c < 0, then ac > bcif a > b and if c > 0, then ac > bcif a > b and if c < 0, then ac < bc
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Examples:Solve each inequality and graph the solution set.
1. 4x + 7 > 2x - 3
2. 2 - 3x < 5
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Examples:Solve the combined inequality and graph the solution set.
1. -5 < 3x - 2 < 1
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Inequalities Involving Absolute Value
If a is any positive number and if u is any algebraic expression, then
|u| < a is equivalent to -a < u < a
|u| < a is equivalent to -a < u < a
Example: Solve the inequality |2x + 4| < 3 and graph.
-3 < 2x + 4 < 31 02345 1 2 3 4 5
A.8 Interval Notation and Solving Inequalities 2010
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Inequalities Involving Absolute Value
If a is any positive number and if u is any algebraic expression, then
|u| > a is equivalent to u < -a or u > a
|u| > a is equivalent to u < -a or u > a
Example: Solve the inequality |2x - 5| > 3 and graph.
2x - 5 < -3 or 2x - 5 > 31 02345 1 2 3 4 5
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Sep 13:04 PM
Homework:page 1028 ﴾6 9, 11 16, 54 56, 59, 61, 63, 66, 69, 73, 77, 91, 97﴿