A.8 Interval Notation; Solving Inequalities

[Pages:10]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

Write each interval as an inequality involving x.

5. [1, 4)

6. (2, )

7. [2, 3]

8. (-, -3]

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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

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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 < 3

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