Absolute Value Inequalities

1. Absolute Value Inequalities

2. > Greater Than is an “or” Statement Graphs go different directions. Thumbs out <---|---|---|---|---|---|---|---|---|---> < Less Than is an “and” Statement Graphs must overlap Thumbs in <---|---|---|---|---|---|---|---|---|---> G. O. L. A.

3. Remember, ‘and’ statements can be written as a compound sentence X > -1 and X < 5 Turn this one around Bring this one down Drop the word “and” NOW PUT THEM TOGETHER

4. Now what 2 numbers will give us this 2? 1) | x + 4 | ≤ 2 L. A. Less = And Now find the value of “x”, set up 2 inequalities using the information found between the absolute value bars….. BUT FIRST – what happens to the inequality symbol when you multiply or divide by a negative number? You flip it And one for the POSITIVE DON’T FLIP!!! One for the NEGATIVE Don’t forget to flip the symbol so... Now put them together

5. Let’s graph-6 ≤ x ≤-2 What kind of Circles? Now shade Between the circles !! <---|---|---|---|---|---|---|---|---|---> -8 -7-6 -5 -4 -3 -2 -1 0 When graphing COMPOUND INEQUALITIES ( the AND statements ), shading stops at the endpoints. Another way to say it: X is between -6 and -2 So…. Shade between -6 and -2

6. What kind of statement? 2) | x - 9 | ≤ 5 Now what 2 numbers will give us this 5? less than is an “and” statement thumbs in to graph Set up the NEGATIVE side Don’t forget to flip the symbol And set up the POSITIVE side DON’T FLIP!!! so... Now put them together

7. Let’s graph it.. 4 ≤ x ≤14 Now shade Between the circles !! What kind of Circles? <---|---|---|---|---|---|---|---|---|---> 2 4 6 8 10 12 14 16 18 When graphing COMPOUND INEQUALITIES ( the AND statements ), shading stops at the endpoints. Another way to say it: X is between 4 and 14 So…. Shade between 4 and 14

8. What kind of statement? 3) | 4x – 5 | < 3 You try it! less than is an “and” statement – thumbs in Set up the NEGATIVE side Don’t forget to flip the symbol And set up the POSITIVE side DON’T FLIP!!! so... Now put them together

9. Graph it ½ < x < 2 Now shade Between the circles !! What kind of Circles? <---|---|---|---|---|---|---|---|---|---> 0 1 2 3 When graphing COMPOUND INEQUALITIES ( the AND statements ), shading stops at the endpoints. Another way to say it: X is between ½ and 2 So…. Shade between ½ and 2

10. What about this one? What kind of statement? 4) | x | > 4 greater than is an “or” statement Remember, “or” in opposite directions – Thumbs out Set up the NEGATIVE side Don’t forget to flip the symbol And set up the POSITIVE side DON’T FLIP!!! Or’s are not compound statements! Don’t put them together. so... Graph <---|---|---|---|---|---|---|---|---|---> -4 -2 0 2 4

11. You try it 5) | 2x + 1 | ≥ 8 What kind of statement? greater than is an “or” statement - Thumbs out Set up the NEGATIVE side Don’t forget to flip the symbol And set up the POSITIVE side DON’T FLIP!!! so...

12. Now graph it! • <---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---> -5 -4 -3 -2 -1 0 1 2 3 4 5 6 "or" - thumbs out

13. Your turn What kind of statement? 6) | x + 9 | ≥ 7 Set up the NEGATIVE side Don’t forget to flip the symbol And set up the POSITIVE side DON’T FLIP!!! so... Now graph it.. • <---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---> -16 -14 -12 -10 -8 -6 -4 -2 0 2

14. 7) | x + 9 | ≥ -5 • The absolute value of any number is positive, therefore | x + 9 | ≥ -5 + ≥ - A POSITIVE IS ALWAYS GREATER THANA NEGATIVETherefore ALL REAL NUMBERS Is the answer!!! Absolute Value Is Pos. True

15. 8) | x - 6 | ≤ -2 • The absolute value of any number is positive, therefore | x - 6 | ≤ -2 + ≤ - A POSITIVE IS NEVER LESS THANA NEGATIVETherefore NO SOLUTION Is the answer!!! Absolute Value Is Pos. False