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Absolute Value Inequalities G. O. L. A. > Greater Than is an “or” Statement < Less Than is an “and” Statement Graphs go different directions. Thumbs out <---|---|---|---|---|---|---|---|---|---> Graphs must overlap Thumbs in <---|---|---|---|---|---|---|---|---|---> Remember, ‘and’ statements can be written as a compound sentence X > -1 and X < 5 Bring this one down Turn this one around Drop the word “and” 1 x x 5 NOW PUT THEM TOGETHER 1 x 5 1) | x + 4 | ≤ 2 Now what 2 numbers will give us this 2? 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? One for the NEGATIVE And one for the POSITIVE Don’t forget to flip the symbol DON’T FLIP!!! x 4 2 4 4 x 6 Now put them together x4 2 4 4 x 2 x 6 and x 2 6 x 2 Let’s graph -6 ≤ x ≤-2 <---|---|---|---|---|---|---|---|---|---> -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 What kind of statement? 2) | x - 9 | ≤ 5 less than is an “and” statement thumbs in to graph Now what 2 numbers will give us this 5? Set up the NEGATIVE side And set up the POSITIVE side Don’t forget to flip the symbol DON’T FLIP!!! x 9 5 9 9 x 4 Now put them together x 9 5 9 9 x 14 x 4 and x 14 4 x 14 Let’s graph it.. 4 ≤ x ≤14 <---|---|---|---|---|---|---|---|---|---> 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 3) | 4x – 5 | < 3 What kind of statement? less than is an “and” statement – thumbs in Set up the NEGATIVE side And set up the POSITIVE side Don’t forget to flip the symbol DON’T FLIP!!! 4 x 5 3 5 5 4x 2 4 4 4x 5 3 5 5 4x 8 4 4 2 1 x or x 4 2 Now 1 put them x and x 2 together 2 x2 1 x2 2 Graph it ½<x<2 <---|---|---|---|---|---|---|---|---|---> 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 4) What kind of statement? |x|>4 greater than is an “or” statement Remember, “or” in opposite directions – Thumbs out Set up the NEGATIVE side And set up the POSITIVE side Don’t forget to flip the symbol DON’T FLIP!!! x4 x 4 x 4 or x 4 Graph <---|---|---|---|---|---|---|---|---|---> -4 -2 0 2 4 Or’s are not compound statements! Don’t put them together. 5) | 2x + 1 | ≥ 8 greater than is an “or” statement - Thumbs What kind of statement? out Set up the NEGATIVE side And set up the POSITIVE side Don’t forget to flip the symbol DON’T FLIP!!! 2x 1 8 1 1 2x 9 2 1 9 2 x 2 or x 4 2 1 x 4 2 2x 1 8 1 1 2x 7 2 2 1 7 x or x 3 2 2 1 or x 3 2 1 1 x 4 or x 3 2 2 <---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---> -5 -4 -3 -2 -1 0 1 2 3 4 5 6 What kind of statement? 6) | x + 9 | ≥ 7 Set up the NEGATIVE side And set up the POSITIVE side Don’t forget to flip the symbol DON’T FLIP!!! x9 7 9 9 x 2 x 9 7 9 9 x 16 x 16 or x 2 <---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---> -16 -14 -12 -10 -8 -6 -4 -2 0 2 7) | x + 9 | ≥ -5 Absolut e Value Is Pos. The absolute value of any number is positive, therefore | x + 9 | ≥ -5 +≥ - 8) | x - 6 | ≤ -2 Absolut e Value Is Pos. The absolute value of any number is positive, therefore | x - 6 | ≤ -2 +≤ -