2 5 Inequalities Solving inequalities follow all the same rules as solving equations with one exception When multiplying or dividing both sides of an inequality by a negative number the inequality sign is reversed that is if c is a negative number and a b then ac bc and Solve and graph express solution set in interval notation 1 9x 36 2 8x 11 3x 13 3 8x 3 3 2x 1 x 5 4 x 14 2 x 1 x 3 3 2 2 5 3 y 8 2 10 y 5 y 1 a b c c If two simple inequalities are joined by the connective and or or we have a compound inequality The connective and indicates that both simple inequalities must be true for the compound inequality to be true A compound inequality using or is true if either one or both of the simple inequalities is true When graphing compound inequalities and means to shade the points that the two simple inequalities have in common this is called the intersection In symbols x x 2 and x 5 x x 2 x x 5 The graph would be the points between 2 and 5 or 2 5 The connective or means to shade all the points that are true for either of the simple inequalities this is called the union In symbols x x 2 or x 5 x x 2 x x 5 The graph is the entire number line because all points are either greater than 2 or less than 5 or Inequalities may be read from left to right or right to left you should read from the variable A compound inequality such as x 1 and x 5 is usually written as 1 x 5 This is an inequality with three members 1 x and 5 To solve a compound inequality connected by or solve each simple inequality individually For the 3 member and compound inequality isolate the variable by applying the rules for simple inequalities to all 3 members simultaneously Solve and graph each compound inequality 1 1 1 1 x or x 2 6 4 3 5 2 60 15x 20 12 2 x 4 15x 8 x 8 15 or x 4 7 1 3 2x 11 4 2x 8 2 x 4 8 3 2x 1 7 3 9 2x 1 21 8 2x 22 4 x 11 9 An analyst has determined that the cost to produce and sell x units of a certain product is C 70x 8000 Revenue for the product is R 120x Find the values of x for which the company will at least break even R C 120x 70x 8000 The company will breakeven if it sells 160 units 50x 8000 It will make a profit if it sells more than 160 units x 160