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Section 3-1 Inequalities and their Graphs SPI 22N: identify the graphical representation of the solution to a one variable inequality on a number line Objectives: • Identify solutions of inequalities • Graph and write inequalities Vocabulary Solution of an inequality: • any number that makes the equation true. Example: For the inequality x > 3, all numbers that are less than 3 make the inequality true. Remember the Signs of Inequality > Greater than < Less than = Equal to Less than and equal to Greater than and equal to Practice Understanding Inequalities Is each number a solution of x > 5? a. –2 No, –2 > 5 is not true. b. 10 c. 25 5 Yes, 10 > 5 is true. Yes, 5 > 5 is true. Is each number a solution of 3 + 2x < 8? a. –2 3 + 2x < 8 b. 3 3 + 2x < 8 3 + 2(–2) < 8 Substitute for x. 3 + 2(3) < 8 Substitute for x. 3–4<8 Simplify. 3+6<8 Simplify. –1 < 8 Compare. 9<8 Compare. –2 is a solution. 3 is not a solution. Representing Inequalities on a Number Line An inequality may have more than one solution. Use the following symbols to graphically represent solutions to an inequality. Closed dot on a number line shows solution includes value Open dot on a number line shows solution does not include value a. Graph d < 3. The solutions of d < 3 are all the points to the left of 3. b. Graph –3 ≥ g. The solutions of –3 > g are –3 and all the points to the left of –3. Writing Inequalities from Number Lines x<2 Numbers less than 2 are graphed. x < –3 Numbers less than or equal to –3 are graphed. x > –2 x > 1 2 Numbers greater than –2 are graphed. Numbers greater than or equal to 1 2 are graphed. Write an Inequality for each Situation a. A speed that violates the law when the speed limit is 55 miles per hour. Let v = an illegal speed. The speed limit is 55, so v > 55. b. A job that pays at least $500 a month. Let p = pay per month. The job pays $500 or more, so p > 500. How would you graph the solutions to the above problems on a number line? Real World: Using Inequalities Suppose your school plans a musical. The goal is to have ticket sales of at least $4000. Adult tickets are $5.oo and student tickets are $4.00. Let a represent the number of adult tickets and s represent the number of student tickets. Write an inequality that represents the school’s goal. 5a + 4s 4000