OBJECTIVE I will use inverse operations to solve inequalities with the four basic operations.

Vocabulary Graph of Linear Inequality - set of all points on a number line that represent all solutions of an inequality Equivalent Inequalities - two or more inequalities that have the same solution

Inequalities x < 5 x > -2 x ≤ 3 x ≥ 4

Writing and Graphing an Inequality The freezing point of water is 32ºF. At temperatures at or below the freezing point, water is ice. Write an inequality that gives the temperatures at which water is ice. Then graph the inequality.

Writing and Graphing an Inequality Solution Let t be the temperature of the water. t ≤ 32

Subtraction Property of Inequality Subtracting the same number on each side of an inequality produces an equivalent inequality. If a < b, then a - c < b - c If a > b, then a - c > b - c

Solving an Inequality Using Subtraction x + 7 ≥ 16Original Inequality Subtract 7 from both sides x ≥ 9 Simplify Graph

Addition Property of Inequality Adding the same number on each side of an inequality produces an equivalent inequality. If a < b, then a + c < b + c If a > b, then a + c > b + c

Solving an Inequality Using Addition x - 9 ≥ 8Original Inequality Add 9 to both sides x ≥ 17Simplify Graph

Guided Practice 1.x + 19 < 12 2.x - 13 < -5

Writing and Solving an Inequality Biathlon You are competing in a biathlon, a sports competition with two events. Last year, you finished the biathlon in 47 minutes. If you get 31 minutes in the swimming event this year, what are the times you can post in the running event in order to beat last year’s finishing time?

Writing and Solving an Inequality Solution Let t represent this year’s running time. Swimming + Running < Last year’s time 31 + t < 47 t < 16 Answer You must post a time better than 16 minutes to beat last year’s time.

Dividing Two Sides of an Inequality Positive 8 < 16 8 ÷ 2 < 16 ÷ 2Divide both sides by constant 2 4 < 8 Simplify Negative 8 < 16 8 ÷ (-2) < 16 ÷ (-2)Divide both sides by constant < -8XSimplify -4 > -8 Flip inequality sign

Division Property of Inequality Dividing each side of an inequality by a positive number produces an equivalent inequality. If a 0, then a ÷ c < b ÷ c. Dividing each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. If a b ÷ c.

Solving an Inequality Using Division Original Inequality Divide each side by 4 Simplify Graph

Solving an Inequality Using Division Original Inequality Divide each side by -7 Reverse inequality symbol Simplify Graph

Guided Practice

Writing and Solving an Inequality The city of Topock floods when there is 18 inches of rain in one storm. If it is raining so that 3 inches of rain fall per hour, how long can it rain before the city floods. Rate of rain Hours of rain < Flooding point 3x < 18Substitute x < 6Divide both sides by 3 It has to rain less than six hours so that Topock does not flood.

Multiplying Two Sides of an Inequality Positive 2 < < 3 4Multiply both sides by constant 3 6 < 12 √Simplify Negative 2 < < -3 4Multiply both sides by constant < -12XSimplify -6 > -12 √Flip inequality sign

Multiplication Property of Inequality Multiplying each side of an inequality by a positive number produces an equivalent inequality. If a 0, then ac < bc. Multiplying each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. If a bc.

Solving an Inequality Using Multiplication Original Inequality Multiply each side by 7 Simplify Graph

Solving an Inequality Using Multiplication Original Inequality Multiply each side by -6 Reverse inequality symbol Simplify Graph

Guided Practice

Independent Practice 1.Milk boils at 218ºF. At temperatures at or above its boiling point, milk boils. Write an inequality to represent the temperatures at which milk boils. Then graph the inequality. 2.x + 9 < 6 3.x - 4 > If a double bicycle can hold a maximum of 275 pounds and one person weighs 132 pounds, what are the possible weights for the seconds person?