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Lesson 1 – 5 Solving Inequalities
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Objectives Solve inequalities.
Solve real-world problems involving inequalities.
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Trichotomy Property For any two real numbers, a and b, exactly one of the following statements is true.
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Addition Property of Inequality
For any real numbers a, b, and c:
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Subtraction Property of Inequality
For any real numbers a, b, and c:
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Solution Sets These properties can be used to solve inequalities. The solution sets of inequalities in one variable can then be graphed on number lines. Use a circle with an arrow to the left for < and an arrow to the right for >. Use a dot with an arrow to the left for ≤ and an arrow to the right for ≥.
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Example 1 {x | x > 9} Subtract 6x from each side.
Add 5 to each side. We must write our answers in set notation. {x | x > 9} 7 8 9
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Multiplication Property of Inequality
For any real numbers a, b, and c: If c is positive: If c is negative:
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Division Property of Inequality
For any real numbers a, b, and c: If c is positive: If c is negative:
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Example 2 {y | y ≤ -8} Divide each side by -0.25.
Remember: When multiplying or dividing an inequality by a negative number, you must change the direction of the inequality. {y | y ≤ -8} -10 -9 -8
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Interval Notation The solution set of an inequality can also be described by using interval notation. The infinity symbols are used to indicate that a set is unbounded in the positive or negative direction, respectively. To indicate that an endpoint is not included is the set, parentheses are used.
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Interval Notation Example 3
Write the solution set to the number line below using interval notation. 1 2
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Interval Notation Example 4
Write the solution set to the number line below using interval notation. -2 -1
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Example 5 Multiply each side by 9. Subtract m from each side.
Divide each side by -10. -1
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Example 6 Distribute on each side. Simplify.
Subtract g from each side. Subtract 8 from each side. 1 2
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Example 7 Is this true? Cross Multiply. Distribute on each side.
Subtract 12x from each side. Is this true?
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“Or” Compound Inequalities
The graph of a compound inequality containing or is the union of the solution sets of the two inequalities.
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Example 8 1 4
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Example 9 Solve each inequality separately. Now graph the union. -7 -1
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“And” Compound Inequalities
A compound inequality containing the word and is true if and only if both inequalities are true.
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Example 10 Graph x ≥ -1 and x < 2. x ≥ -1 x < 2 -1 ≤ x < 2
1 2 x < 2 -1 1 2 -1 ≤ x < 2 -1 1 2 The final graph is the intersection of the first two graphs.
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Example 11 Subtract 7 from all “three” sides. Divide everything by 2.
3 5 You can also rewrite this as 13 < 2x + 7 and 2x + 7 ≤ 17.
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