Inequalities (Multi Step & Compound) 09-01-15

Some inequalities have variable terms on both sides of the inequality symbol (just like equations). You can solve these inequalities like you solved equations with variables on both sides.

Example 1 Solve the inequality and graph the solutions. y ≤ 4y + 18 Subtract 4y from both sides. ___ ___ -3 -3 Since y is multiplied by -3, divide both sides by -3 to undo the multiplication. y  –6 Since you divided by a negative, you must flip the inequality sign. –10 –8 –6 –4 –2 2 4 6 8 10

Example 5 Solve the inequality and graph the solutions. –12 –9 –6 –3 3 Solve the inequality and graph the solutions. 2(k – 3) > 6 + 3k – 3 Distribute 2 on the left side of the inequality. And, combine the 6 and -3 on the right side. 2(k – 3) > 3 + 3k 2k + 2(–3) > 3 + 3k 2k – 6 > 3 + 3k Subtract 3k from both sides. –3k – 3k -k – 6 > 3 Since 6 is subtracted from k, add 6 to both sides to undo the subtraction. +6 +6 -k > 9 Divide both sides by -1 (flip the inequality). k < -9

There are special cases of inequalities (just like equations) called identities and contradictions.

Compound Inequalities The inequalities we have done so far are simple inequalities. When two simple inequalities are combined they become compound. Two inequalities that are joined by the word and or the word or

AND This means that we are looking for where the solutions overlap – the intersection. The answer must be a solution to BOTH inequalities. x > 8 and x < 12 This would be all real numbers that are larger than 8 and 12 or less.

AND

OR This means to take the two solution sets and combine them. The answer will be any number that is in EITHER set. x > 11 or x < 7

OR

Example 1 Solve: 3x – 8 < 7 and 2x + 1 > 5 First, solve each equation 3x < 15 x < 5 2x > 4 x > 2 Answer is x < 5 and x > 2

Example 1 (cont) When your answer has AND you must write a repeated inequality Put the lower number on the left, the higher number in the right and the variable in the middle. Adjust the signs x < 5 and x > 2 would be 2 < x < 5 2 4 5

Example 2 1 10 Solve 3x + 8 < 11 or -2x + 4 < -16 Solve each inequality 3x < 3 x < 1 -2x < -20 x > 10 x < 1 or x > 10 1 10

Example 3 5 < x + 3 < 12 This means 5 < x + 3 and x + 3 < 12 So, solve each equation. 2 < x and x > 9 Combine 2 < x < 9 Graph 2 7 9

An answer like x > 5 and x < 3 would become 3 < x > 5 But, if I try to graph it, it will look like this. Where does it overlap? It has to overlap if it is “and” Since it does not overlap, the solution would be “No Solution” 3 4 5