Section P.2 Solving Inequalities 1.Solutions of inequalities are all values that make it true (or satisfy the inequality); called a solution set Bounded solution sets: do not have infinity as one of the numbers Ex. (-3,5] corresponds to –3<x<5 [0,2] corresponds to 0<x<2 (graph) Unbounded solution sets: have infinity for at least one of the numbers Ex. (-3,∞) corresponds to –3<x<∞ (graph) (-∞, ∞) corresponds to -∞<x<∞ (graph) (which is also the set of all real numbers R)

2. Properties of Inequalities when multiplying or dividing by a negative number you must reverse the inequality sign(s) Transitive property: If a < b and b < c  a < c Addition of Inequalities: If a < b and c < d  a+c < b+d Addition of a Constant: If a < b  a+c < b+c Multiplication by a Constant: For c > 0 and a < b  ac < bc For c bc

3. Linear Inequalities in one variable Ex. 5x-7 > 3x+9 (show graph and interval notation) Ex. 1-3x > x-4 (show graph and interval notation) 2 double inequality: you want to isolate x in the middle Ex. –4 < 5x-2 < 7 note: if you need to reverse the sign you must reverse both signs!!

4. Absolute Value Inequalities (rule #4) if |x| < a, then the values of x lie between –a and a Ex. |x-5| < 2 note: for less than absolute value inequalities think “trap it” if |x| > a, then the values of x are less than –a and greater than a Ex. |x+3| > 7 note: for greater than absolute value inequalities think “separate it”

5. Polynomial Inequalities Factor to find critical values, then test a point in each interval Ex. x²-x-6 < 0 (chart, graph and interval notation) Ex. 2x-7 < 3 (chart, graph and interval notation) x-5 note: you must make a chart each time!