Chapter 12 Section 2

1. Chapter 12 Section 2 Solving Addition and Subtraction Inequalities

2. Addition and Subtraction Properties for Inequalities Words: For any inequality, if the same quantity is added or subtracted to each side, the resulting inequality is true. Symbols: For all numbers a, b, and c, 1. If a › b, then a + c › b + c and a – c › b – c. 2. If a ‹ b, then a +c ‹ b +c and a – c ‹ b – c. Numbers: -5 ‹ 1 -5 + 2 ‹ 1 + 2 -3 ‹ 3 2 › -4 2 – 3 › -4 – 3 -1 › 7

3. Example 1 Solve x + 14 ≥ 5 x + 14 ≥ 5 - 14 -14 x ≥ -9 The solution is all numbers greater than or equal to -9 . To check your answer substitute a number less than -9 and greater into the inequality.

4. Your Turn Solve the inequality. Check your Solution. x + 2 ‹ 7 All numbers less than 5

5. Your Turn Solve the inequality. Check your Solution. x - 6 ≥ 12 All numbers greater than or equal to 18

6. Set-builder Notation A more concise way to express the solution to an inequality is to use set-builder notation. The solution in example 1 in set-builder notation is {x l x ≥ -9}. {x l x ≥ -9} The set of all numbers x Such that X is greater than or equal to -9

7. In lesson 12-1, you learned that you can show the solution to an inequality on a line graph. The solution, {x l x ≥ -9}, is shown below. -12 -11 -10 -9 -8 -7

8. Example 2 Solve 7y + 4 › 8y -12. Graph the solution. 7y + 4 › 8y -12 7y – 7y + 4 › 8y – 7y -12 4 + 12 › y-12 + 12 16 › y Since 16 › y is the same as y ‹ 16, the solution is {y l y ‹ 16}. ` 14 15 16 17 18 The graph of the solution has a circle at 16, since 16 is not included. The arrow points to the left.

9. Your Turn Solve each inequality. Graph the solution. 5y - 3 › 6y - 9 {y l y ‹ 6 } 4 5 6 7 8

10. Your Turn Solve each inequality. Graph the solution. 3r + 7 ≤ 2r + 4 {y l y ≤ -3 } -5-4 -3 -2 -1