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Goals: Solving Inequalities 1. Use graphs to represent the solution set of an inequality. 2. Use set-builder notation to represent solution set of an inequality. 3. Apply addition and multiplication principles to solving inequalities. A solution to an inequality is a value of the variable that makes the inequality true. Inequalities relate variables and numbers with greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤) symbols. Examples: x = -5, x = 2, x = 5 are all solutions to the inequality 5 ≥ x, but x = 5 is not a solution to the inequality 5 > x. Number lines are often used to represent all solutions to an inequality. Represents the solutions to 5 ≥ x. The filled-in point at 5 means include 5 in the solution set. This represents the solutions to x < 5. The open circle at x = 5 indicates that x = 5 is not in the solution set. Open circle – exclude that point. Closed circle – include that point. 1 Addition Principle for Inequalities: For any real numbers a, b, and c; Equivalent inequalities: Two inequalities with the same solution set are said to be equivalent. a < b is equivalent to a + c < b + c a ≤ b is equivalent to a + c ≤ b + c a > b is equivalent to a + c > b + c a ≥ b is equivalent to a + c ≥ b + c Same as with equalities. Example: x + 7 > 12 is equivalent to x > 5 because x + 7 > 12 −7 −7 x > 5 Multiplication Principle for Inequalities: For any real numbers a and b, and for any positive number c: a < b is equivalent to ac < bc and a > b is equivalent to ac > bc. But, if c is negative a < b is equivalent to ac > bc and a > b is equivalent to ac < bc. Multiplying by a negative flips the inequality. The same is true for ≥ and ≤. Why does multiplying by a negative flip the inequality? Examples: 1. 2x < 6 We know that 5 < 7. The multiplication principal tells us that . 5 2< 7. 2 or 10 < 14. 2. − 1 x≥9 3 But is we multiply both side by –2, we get –10 > –14; Only true because we flipped the inequality. 3. – 5x + 2 > –8 Check! 2 Set-builder notation. The set x < 2 can be written {x | x < 2} which is read, “the set of all x such that x is less than 2.” 3