A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
A.8 Interval Notation; Solving Inequalities
Objective:
• Use interval notation
• Use properties of inequalities
• Solve Linear Inequalities
• Solve Combined Inequalities
• Solve Absolute Value Inequalities
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A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
Interval Notation
Let 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
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Open Interval: written (a, b), consists of all real numbers x for which a < x < b
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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
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A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
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, ∞)
x>a
2. (a, ∞) x > a
3. (-∞, a] x < a
4. (-∞, a) x < a
5. (-∞, ∞) x = R
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A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
Examples:
Write each inequality using interval notation.
1. 1 < x < 3
2. -4 < x < 0
3. x > 5
4. x < 1
7. [2, 3]
8. (-∞, -3]
Write each interval as an inequality involving x.
5. [1, 4)
6. (2, ∞)
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A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
Properties of Inequalities
Nonnegative 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 + c
if 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 < 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
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A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
Examples:
Solve each inequality and graph the solution set.
1. 4x + 7 > 2x - 3
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2. 2 - 3x < 5
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A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
Examples:
Solve the combined inequality and graph
the solution set.
1. -5 < 3x - 2 < 1
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A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
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 < 3
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A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
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 > 3
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A.8 Interval Notation and Solving Inequalities 2010
September 22, 2010
Homework:
page 1028 ﴾6 ­ 9, 11 ­ 16, 54 ­ 56, 59, 61, 63, 66, 69, 73, 77, 91, 97﴿ Sep 1­3:04 PM
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