Inequalities and Applications Section 9.1 Inequalities and Applications

Recall Solve and check the equation 3x -2 (x + 3) = 2(x - 3)

Properties to Solve Inequalities The Addition Principle for Inequalities For any real number a, b, and c a < b is equivalent to a + c < b + c a > b is equivalent to a + c > b + c The Multiplication Principle for Inequalities For any real numbers a, b, and for any positive number c, a < b is equivalent to ac < bc a > b is equivalent to ac > bc For any real numbers a, b, and for any negative number c, a < b is equivalent to ac > bc a > b is equivalent to ac < bc

Types of answer No Solution All Real Solution Group of Solutions The variable cancels out and you have a false statement All Real Solution The variable cancels out and you have a true statement Group of Solutions The variable does not cancel out and you have three ways to write the answer

How to Write the Answer Set Builder Notation Interval Notation {variable | interval } Interval Notation [included, included], [included, not included), (not, included, included], (not included, not included) Graphing Notation Use open circle or parentheses to not include the value Use filled in circle or bracket to include the value

Example Solve the inequality and write answer in set builder notation 2x – 5 ≥ 9

Example Solve the inequality and write answer in interval notation form. 8 – x < 15

Example Solve the inequality and write answer in graphing notation 2(4 + 2x) > 2x + 3(2 - 5x)

Example Solve the inequality and write the answer in all three forms. 3(x – 2) + 3 < 2(x – 1) + x

Example Solve the inequality and write the answer in all three forms. 3x – 2 < 2(x – 1) + x

Word Problem Jenn can rent a moving truck for either $99 with unlimited mileage or $49 plus $0.80 per mile. For what mileages would the unlimited mileage plan save money?

Homework 9.1 11, 17, 19, 21, 23, 39, 41, 45