Solving Inequalities Using Addition and Subtraction

1. Solving Inequalities Using Addition and Subtraction Lessons 3-1 and 3-2

2. Addition Property of Inequalities – If any number is ________________ to each side of a true ___________________, the resulting inequality is also ________________. Example A 3 -5 3 + 2 -5 + 2 _____ _____ added equation true > > 5 > -3

3. Example B n – 12 < 65 n – 12 +12 < 65 + 12 Work inequalities horizontally. n < 77 This is called “set-builder notation.” It would be read as “n such that n is less than 77.” { n│ n < 77} “Open” circle (unshaded) at 77, then shade to the left (because it is “less than”). This means any number smaller than 77 is a solution to the inequality

4. Example C k – 4 > 10 Adding the same number to each side of an inequality does not change the direction of the inequality. k – 4 + 4 > 10 + 4 k > 14 Set builder notation is always placed inside of braces. {k │k > 14 }

5. Graphing on the Number Line (A Quick Review) Great than or equal to (≥) and less than or equal to (≤) uses a filled in (or closed) circle then shade the line in the same direction the symbol is pointing. Great than (>) and less than (<) uses an unshaded (or open) circle then shade the line in the same direction the symbol is pointing.

6. 12 + 9 ≥ y – 9 + 9 Add y 21 ≥ y 21 ≤ y│y≤ 21

7. Subtraction Property of Inequality – If any number is ___________ from ________ side of a true inequality, the resulting inequality is also _______. subtracted each true q + 23 - 23 14 - 23 < subtract -9 q < {q│q < –9}

8. m + 15 – 15 ≤ 13 - 15 x – 2 < 8 x – 2 + 2 < 8 + 2 m ≤ –2 { m │ m ≤ –2 } x < 10 { x │ x < 10} The symmetric property does not work for inequalities, so if you “turn the inequality around” you have to change the sign, too.

9. Variables on Both Sides Example H 12n – 4 ≤ 13n 12n –12n – 4 ≤ 13n –12n – 4 ≤ n n ≥ – 4 { n │n ≥ – 4}

10. Example I 3p – 6 ≥ 4p 3p – 3p – 6 ≥ 4p – 3p – 6 ≥ p p ≤ – 6 { p │ p ≤ – 6 }

11. Example J 5x + 4 > 4x + 10 5x + 4 – 4x > 4x – 4x + 10 x + 4 > 10 x + 4 – 4 > 10 – 4 x > 6 {x│x > 6 }

12. Ex. K Seven time a number is greater than 6 times that number minus two. Ex. L Three times a number is less than two times that number plus 5. 3x < 2x + 5 7x > 6x – 2 3x – 2x < 2x – 2x+ 5 7x – 6x > 6x – 6x – 2 x < 5 x > – 2