Compound Inequalities

Compound Inequalities 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 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

Look at page 37 in your workbook. The diagram shows the number of boxes of oranges that an orange tree can produce in 1 year. An grower earns $9.50 for each box of oranges that he sells. How much can the grower expect to earn in 1 year from 1 tree? Explain your reasoning.

What’s another representation you could use to present your solution? All real numbers greater than or equal to -4 and less than 6.

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 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

Try these… 4x + 3 < -5 or –2x + 7 < 1 -3 < j + 2 < 7 (use “and” here) -5 < 3x + 1 < 7

Try these…answers 4x + 3 < -5 or –2x + 7 < 1 -2 3 4x + 3 < -5 or –2x + 7 < 1 -3 < j + 2 < 7 (use “and” here) -5 < 3x + 1 < 7 -5 5 -2 2