Absolute Value Inequalities

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

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

Don’t forget to flip the symbol And one for the POSITIVE Now what 2 numbers will give us this 2? L. A. Less = And 1) | x + 4 | ≤ 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? You flip it One for the NEGATIVE Don’t forget to flip the symbol And one for the POSITIVE DON’T FLIP!!! so... Now put them together

Let’s graph -6 ≤ x ≤-2 <---|---|---|---|---|---|---|---|---|---> What kind of Circles? Now shade Between the circles !! 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

2) | x - 9 | ≤ 5 so... less than is an “and” statement 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

What kind of Circles? Now shade Between the circles !! 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

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

Now shade Between the circles !! What kind of Circles? 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

What about this one? so... 4) | x | > 4 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

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...

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

Your turn so... Now graph it.. -16 -14 -12 -10 -8 -6 -4 -2 0 2 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

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

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