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Lesson 1 – 5 Solving Inequalities.

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1 Lesson 1 – 5 Solving Inequalities

2 Objectives Solve inequalities.
Solve real-world problems involving inequalities.

3 Trichotomy Property For any two real numbers, a and b, exactly one of the following statements is true.

4 Addition Property of Inequality
For any real numbers a, b, and c:

5 Subtraction Property of Inequality
For any real numbers a, b, and c:

6 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 ≥.

7 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

8 Multiplication Property of Inequality
For any real numbers a, b, and c: If c is positive: If c is negative:

9 Division Property of Inequality
For any real numbers a, b, and c: If c is positive: If c is negative:

10 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

11 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.

12 Interval Notation Example 3
Write the solution set to the number line below using interval notation. 1 2

13 Interval Notation Example 4
Write the solution set to the number line below using interval notation. -2 -1

14 Example 5 Multiply each side by 9. Subtract m from each side.
Divide each side by -10. -1

15 Example 6 Distribute on each side. Simplify.
Subtract g from each side. Subtract 8 from each side. 1 2

16 Example 7 Is this true? Cross Multiply. Distribute on each side.
Subtract 12x from each side. Is this true?

17 “Or” Compound Inequalities
The graph of a compound inequality containing or is the union of the solution sets of the two inequalities.

18 Example 8 1 4

19 Example 9 Solve each inequality separately. Now graph the union. -7 -1

20 “And” Compound Inequalities
A compound inequality containing the word and is true if and only if both inequalities are true.

21 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.

22 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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