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**Solving Addition and Subtraction Inequalities**

Chapter 12 Section 2 Solving Addition and Subtraction Inequalities

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**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 ‹ ‹ ‹ 3 2 › -4 2 – 3 › -4 – 3 -1 › 7

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Example 1 Solve x + 14 ≥ 5 x + 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.

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

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

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**X is greater than or equal to -9**

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

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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

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

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

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

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Solve an inequality using multiplication EXAMPLE 2 < 7< 7 x –6 Write original inequality. Multiply each side by –6. Reverse inequality symbol. x > –42.

Solve an inequality using multiplication EXAMPLE 2 < 7< 7 x –6 Write original inequality. Multiply each side by –6. Reverse inequality symbol. x > –42.

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