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32 Representing Inequalities Inequalities Recall: Word Expression a is less than b a is greater than b a is at least b a is no less than b a is at most b a is no more than b Interpretation a<b a>b a≥b a≥b a≤b a≤b When you solve an inequality you have multiple solutions Because it’s impossible to list all the solutions, we can represent the answer using a few different forms: An inequality statement A graph on a number line Interval Notation Set Builder Notation Graphing an Inequality on a Number Line When you just want greater than or less than, you use an on the graph. When you want (greater than or equal to) or (less than or equal to), you’ll use Examples: 1. x3 2. x 2 3. 1 x 4 Interval notation is another method for writing inequalities. Keep these things in mind: Open Parenthesis: ( ) Closed Parenthesis: [ ] Infinity: ∞ Negative Infinity: - ∞ Union: Used if the value is not included in the inequality. Used if the value is included in the inequality. The upper end of the inequality goes on forever in the positive direction. The lower end of the inequality goes on forever in the negative direction. Used to join two intervals together when there is a break in the graph So…When using Interval Notation we will be using: Square brackets [ ] to include the endpoint. Parenthesis to ( ) to exclude the endpoint. Examples: 4. x3 5. x 2 6. 1 x 4 Set-builder notation is mathematical shorthand for precisely stating all numbers of a specific set that possess a specific property. x 2 x 6 " x are the elements of the set of real numbers, such that, x is between 2 and 6 not including 2” Examples: 7. x3 8. x 2 9. 1 x 4 33 Practice! Verbal Phrase 1. All real numbers less than 2 2. All real numbers greater than −2 3. All real numbers less than or equal to 1 4. All real numbers greater than or equal to 0 5. All numbers great then 1 and less than or equal to 5 6. All number that are either less then -2 or greater then 3 7. All real numbers Inequality Graph Interval Notation Set Notation Try It on Your Own! Fill in the Missing Pieces of the Chart! Verbal Phrase Inequality 8. Ex. All numbers greater than or equal to 1 and less than 4. Graph 1 x 4 Interval Notation [1,4) Set Notation x 1 x 4 9. The numbers between -3 and 3 not including either. 10. 11. ( , 4) x 12. 13. [5, ) x 1 or x 5 14. -4 -3 -2-1 0 1 2 3 4 5 15. x x 2 or x 3 34 Homework: Interval and Set Notation Use interval notation to describe the number(s) graphed on each number line. 1. 2. 3. -6 -4 -2 0 2 4 6 -6 4. -4 -2 0 2 4 6 5. -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 6. -6 -4 -2 0 2 4 6 Describe the intervals using interval notation. Union: the symbol 7. Numbers that… • belong in one set OR another • are brought together We could describe the numbers that belong to the interval (-5, -1] OR the interval (2, 4) through the notation (-5, -1] (2, 4). Its visual representation is below. (-5, -1] (2, 4) -6 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 8. 6 Create a graph that has the following characteristics. 9. (2, ∞) 10. (-4, 0) (2, 5]. -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 11-14: For each number line, write the given set of numbers in interval notation and set notation. 11. 12. Interval: Interval: Set: Set: 13. 14. Interval: Interval: Set: Set: 15-17: For each inequality, (a) graph on a number line, (b) write in interval notation, and (c) write in set notation. 15. x 4 16. x 1 or x 4 Graph: Graph: Interval: Interval: Set: Set: 17. all real numbers from 23 to 4 , including 23 but not including 4 . Graph: Interval: Set: