Inequalities and Their Graphs Section 3-1 Part 1

Goals Goal To write and identify solutions of inequalities.

Vocabulary Inequality Solution of an inequality

≤ < > ≥ ≠ Definition Inequality - is a statement that two quantities are not equal. The quantities are compared by using the following signs: ≤ A ≤ B A is less than or equal to B. < A < B than B. > A > B A is greater ≥ A ≥ B ≠ A ≠ B A is not

Writing Inequalities < > ≤ ≥ Symbol Meaning Word Phrases is less than Fewer than, below is greater than More than, above is less than or equal to At most, no more than is greater than or equal to At least, no less than

Example: Translating Words into Inequalities Write an inequality for each situation. A. There are at least 35 people in the gym. Let p = the number of people in the gym. p ≥ 35 “At least” means greater than or equal to. B. The carton holds at most 12 eggs. Let e = the number of eggs the carton hold. e ≤ 12 “At most” means less than or equal to.

Your Turn: Write an inequality for each situation. A. There are at most 10 gallons of gas in the tank. Let g = the number of gallons of gas. g ≤ 10 “At most” means less than or equal to. B. There are fewer than 10 yards of fabric left. Let y = the yards of fabric. y < 10 “Fewer than” means less than.

Example: Writing Inequalities Write an inequality for each statement. A. A number m multiplied by 5 is less than 25. A number m multiplied by 5 is less than 25. m • 5 < 25 5m < 25 B. The sum of a number y and 16 is no more than 100. The sum of a number y and 16 is no more than100 y + 16 ≤ 100 y + 16 ≤ 100

Your Turn: Write an inequality for each statement. A. A number y plus 14 is greater than 21. A number y plus 14 is greater than 21. y + 14 > 21 y + 14 > 21 B. A number t increased by 7 is more than 11 A number t is increased by 7 is more than 11 t + 7 > 11 t + 7 > 11

Definition Solution of an Inequality - is any value that makes the inequality true. The solutions of the inequality x<5 are all real numbers x that are less than 5. You can evaluate an expression to determine whether a value is a solution of an inequality.

Example: Identifying Solutions of Inequalities Identify solutions of x – 6 ≥ 4 by evaluating. Is each number a solution? Substitute for x, simplify and compare. x –3 9.9 10 10.1 12 x – 6 –9 –6 3.9 4 4.1 6 x – 6 ≥ 4 –9 4 ≥ –6 4 ≥ 3.9 4 ≥ 4 4 ≥ 4.1 4 ≥ 6 4 ≥ Solution? No No No Yes Yes Yes If the inequality is a true statement, then the value of x is a solution. If the inequality is a false statement, then the value of x is not a solution.

Example: Identifying Solutions of Inequalities Identify solutions of 13 – 7y ≤ 6 by evaluating. Is each number a solution? Substitute for y, simplify and compare. y –2 -1 1 2 3 13 – 7y 27 20 13 6 - 1 - 8 13-7y≤6 27 6 ≤ 20 6 ≤ 13 6 ≤ 6 6 ≤ - 1 6 ≤ - 8 6 ≤ Solution? No No No Yes Yes Yes If the inequality is a true statement, then the value of x is a solution. If the inequality is a false statement, then the value of x is not a solution.

Your Turn: Identify solutions of 2p > 8 by evaluating. Is each number a solution? Substitute for p, simplify and compare. p –3 3.9 4 4.1 5 2p –6 7.8 8 8.2 10 7.8 8 > 8 8 > 8.2 8 > 2p > 8 –6 8 > 0 8 > 10 8 > Solution? No No No No Yes Yes If the inequality is a true statement, then the value of x is a solution. If the inequality is a false statement, then the value of x is not a solution.

Your Turn: Identify solutions of 3x - 2 < -1 by evaluating. Is each number a solution? Substitute for x, simplify and compare. Solution? –3 x 3x - 2 -1 1 3 4 3x-2<-1 –11 -5 - 2 1 7 10 -2 -1 < 1 -1 < 7 -1 < –11 -1 < -5 -1 < 10 -1 < Yes Yes Yes No No No If the inequality is a true statement, then the value of x is a solution. If the inequality is a false statement, then the value of x is not a solution.

Joke Time Why do mother kangaroos hate rainy days? Because the kids have to play inside. Why did the chicken go to the middle of the road? To lay it on the line. Why did the chicken cross the basketball court to talk with the ref? Because he was calling all fowls!