Inequalities and Absolute Value Equations and Inequations

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1 Inequalities and Absolute Value Equations and Inequations Fall Math 1010 A no TIE fighter and squinting cat zone. (Math 1010) M / 12

2 Roadmap Notes for solving inequalities. Examples of solving inequalities. Notes for solving absolute value equations and inequalites. Examples of solving absolute value equations and inequalities. Today s lecture will procede with slide notes and then chalkboard examples. (Math 1010) M / 12

3 2.4 - The Symbols for Inequalities less than Example x < 7 less than or equal Example x 4 greater than Example x > 3 greater than or equal Example x or equal means the values for the unknown can include that number. - Inequalties without or equal, that is < and >, are called strict inequalities. (Math 1010) M / 12

4 2.4 - Notations Styles in this section include intervals, inequalty signs, graphs, and set (or set builder) notation. The samples below are equivalent. Example x + 6 < 9 x < 3 Interval: (, 3) Inequality signs: x < 3 Set builder notation: {x x < 3} (I like to use : in place of.) (Math 1010) M / 12

5 2.4 - Operations Operations on all sides of an inequality with expressions are the same for equalities with one exception: Multiplication and division by a negative quantity produces an equivalent inequality with a reversed inequality symbol. Summary: Add/Subtract: a < b a + c < b + c a c < b c Multiply/Divide: positive quantities a < b ac < bc a c < b c, c > 0 Multiply/Divide: negative quantities a < b ac > bc a c > b c, c < 0 Transitivity: When a < b and b < c, then it follows that a < c. (Math 1010) M / 12

6 2.4 - Compound Inequalties - Conjunctive And Conjunctive (and) is used for compound inequalties that have two conditions. Both conditions must be met. Example 1 5 2x and 5 2x < 7 Write this as a double inequalty and solve. (Math 1010) M / 12

7 2.4 - Compound Inequalties - Conjunctive And Conjunctive (and) is used for compound inequalties that have two conditions. Both conditions must be met. Example 1 5 2x and 5 2x < 7 Write this as a double inequalty and solve x < 7 (Math 1010) M / 12

8 2.4 - Compound Inequalties - Conjunctive And Conjunctive (and) is used for compound inequalties that have two conditions. Both conditions must be met. Example 1 5 2x and 5 2x < 7 Write this as a double inequalty and solve x < 7 6 2x < 2 (Math 1010) M / 12

9 2.4 - Compound Inequalties - Conjunctive And Conjunctive (and) is used for compound inequalties that have two conditions. Both conditions must be met. Example 1 5 2x and 5 2x < 7 Write this as a double inequalty and solve x < 7 6 2x < x > 2 2 (Math 1010) M / 12

10 2.4 - Compound Inequalties - Conjunctive And Conjunctive (and) is used for compound inequalties that have two conditions. Both conditions must be met. Example 1 5 2x and 5 2x < 7 Write this as a double inequalty and solve x < 7 6 2x < x > x > 1 1 < x 3 (Math 1010) M / 12

11 Just Say No (Math 1010) M / 12

12 2.4 - Compound Inequalties - Disjunctive Or Disjuctive (or) is used for compound inequalties that have two conditions. Either condition may be met. These inequalities cannot be written as a compound inequality. Example x + 3 < 7 or x + 3 > 14 Solve. (Math 1010) M / 12

13 2.4 - Compound Inequalties - Disjunctive Or Disjuctive (or) is used for compound inequalties that have two conditions. Either condition may be met. These inequalities cannot be written as a compound inequality. Example x + 3 < 7 or x + 3 > 14 Solve. One at a time: x + 3 < 7 x < 10 (Math 1010) M / 12

14 2.4 - Compound Inequalties - Disjunctive Or Disjuctive (or) is used for compound inequalties that have two conditions. Either condition may be met. These inequalities cannot be written as a compound inequality. Example x + 3 < 7 or x + 3 > 14 Solve. One at a time: x + 3 < 7 x < 10 Next one: x + 3 > 14 x > 11 (Math 1010) M / 12

15 2.4 - Compound Inequalties - Disjunctive Or Disjuctive (or) is used for compound inequalties that have two conditions. Either condition may be met. These inequalities cannot be written as a compound inequality. Example x + 3 < 7 or x + 3 > 14 Solve. One at a time: x + 3 < 7 x < 10 Next one: x + 3 > 14 x > 11 Solution: x < 10 or x > 11. (Math 1010) M / 12

16 2.5 - Absolute Value Equations and Inequalties An absolute value equation contains a term with an absolute value expression. It may have no solution, or it may have one or more solutions. Example x = 3.6 x = 3.6 or x = 3.6 Example x = 12 No solution; an absolute value cannot be negative. To solve x = a, a 0 the algebraic expression inside the absolute value symbols x may be a or a. (Math 1010) M / 12

17 2.5 - Solving Equations Simplify each side to have at most one absolute value expression. Write two linear equations - write one side equal to the other side, and then write one side equal to the opposite of the other side. Solve each linear equation one after the other. Check the solutions! It is possible for no solution or one solution. (Math 1010) M / 12

18 2.5 - Solving Inequalities Absolute value inequalties match a conjunctive pair of statements (also, a compound inequality) or a disjunctive pair of statements. (Math 1010) M / 12

19 2.5 - Solving Inequalities Absolute value inequalties match a conjunctive pair of statements (also, a compound inequality) or a disjunctive pair of statements. Example Solutions to x < 2 lie between -2 and 2. That is, 2 < x < 2. (Math 1010) M / 12

20 2.5 - Solving Inequalities Absolute value inequalties match a conjunctive pair of statements (also, a compound inequality) or a disjunctive pair of statements. Example Solutions to x < 2 lie between -2 and 2. That is, 2 < x < 2. Example Solutions to x > 2 lie outside -2 and 2. That is, x < 2 or x > 2. (Math 1010) M / 12

21 Assignment Assignment: For Monday: 1. Read sections 3.1 and Exercises from 2.4, 2.5 due Monday, September Pre-Exam 1 Wednesday, September 18. (Math 1010) M / 12

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