Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 1 Chapter 3 Systems of Linear Equations Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide Linear Inequalities in Two Variables; Systems of Linear Inequalities Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 3 Satisfy, solution, solution set, and solve for an inequality in two variables Definition If an inequality in the two variables x and y becomes a true statement when a is substituted for x and b is substituted for y, we say the ordered pair (a, b) satisfies the inequality and call (a, b) a solution of the inequality. The solution set of an inequality is the set of all solutions of the inequality. We solve the inequality by finding its solution set. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 4 Example: Sketching the Graph of an Inequality Graph Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 5 Solution Begin by sketching a graph of Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 6 Solution Choose a value of x (say, 4) and find several solutions with an x-coordinate of 4. For our equation, if x = 4, then y = 3. So the point (4, 3) is on the line. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 7 Solution For if x = 4, then So, if x = 4, some possible values of y are y = 3.4, y = 4, and y = 5. The points (4, 3.4), (4, 4), and (4, 5) lie above the point (4, 3), which is on the line. The graph on the next slide shows these points. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 8 Solution Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 9 Solution We shade the region that contains all of the points that represent solution of Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 10 Solution We dash the line to indicate that its points are not solutions of the inequality. For example, the point (4, 3) that is on the line does not satisfy the inequality: Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 11 Solution We can draw a graph of the inequality using a graphing calculator, but we have to imagine the equation is drawn as a dashed line. To shade above a line, press and press twice. Next, press as many times as necessary for the triangle shown in the left figure below to appear. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 12 Graph of an Inequality in Two Variables The graph of an inequality of the form y > mx + b is the region above the line y = mx + b. The graph of an inequality of the form y < mx + b is the region below the line y = mx + b. For either inequality, we use a dashed line to show that y = mx + b is not part of the graph. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 13 Graph of an Inequality in Two Variables The graph of an inequality of the form y mx + b is the line y = mx + b and the region above that line. The graph of an inequality of the form y mx + b is the line y = mx + b and the region below that line. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 14 Example: Sketching the Graph of an Inequality Sketch the graph of 2x 3y > 6. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 15 Solution First, isolate y on one side of the inequality: Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 16 Solution The graph of is the region below the line Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 17 Solution To verify, choose a point on our graph, such as (3, 1), and check that it satisfies the inequality: Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 18 Graph of an Inequality in Two Variables Warning We must first isolate y on the left side of a linear inequality before we can determine whether the graph includes the region that is above or below a line. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 19 Example: Sketching the Graphs of Inequalities Sketch the graph of the inequality. 1. y 32. x > 4 Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 20 Solution 1. The graph of y 3 is the horizontal line y = 3 and the region below that line. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 21 Solution 2. Ordered pairs with x-coordinates great than 4 are represented by points that lie to the right of the vertical line x = 4. So, the graph of x > 4 is the region to the right of x = 4. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 22 Solution, solution set, and solve for a system of inequalities in two variables Definition An ordered pair (a, b) is a solution of a system of inequalities in two variables if it satisfies all the inequalities in the system. The solution set of a system is the set of all solutions of the system. We solve a system by finding its solution set. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 23 Solution Set of a System of Inequalities We can find the solution set of a system of inequalities in two variables by locating the intersection of the graphs of all of the inequalities. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 24 Example: Graphing the Solution Set of a System of Inequalities Solve the system Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 25 Solution Sketch the graph of y 2x + 1 and the graph of Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 26 Solution The graph of the solution set of the system is the intersection of the graphs of the inequalities. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 27 Solution We can use a graphing calculator to draw a graph of the solution set of the system, where we imagine the border of the red line is drawn with a dashed line. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 28 Example: Using a System of Inequalities to Make Estimates A persons life expectancy predicts how many remaining years the person will live. The life expectancies of U.S. females at birth and at 20 years are shown in the table on the next slide for various calendar years. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 29 Example: Using a System of Inequalities to Make Estimates Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 30 Example: Using a System of Inequalities to Make Estimates Let L = B(t) be the life expectancy (in years) at birth and L = T(t) be the life expectancy (in years) at age 20 years, both of U.S. females at t years since The linear regression models of B and T are, respectively, B(t) = 0.097t T(t) = 0.075t Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 31 Example: Using a System of Inequalities to Make Estimates 1. Find a system of inequalities that describes the life expectancies of U.S. females from 0 years through 20 years old from 1985 to Graph the solution set of the system of inequalities you found in Problem Estimate the life expectancies of U.S. females from 0 years through 20 years old in 2015. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 32 Solution 1. The life expectancies must be less than or equal to the at-birth life expectancies ( L 0.097t ) and greater than or equal to the life expectancies at age 20 years (L 0.075t ). We are seeking life expectancies for the calendar years from 1985 through 2020: t 5 and t 40. So our system is: L 0.097t L 0.075t t 5 t 40 Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 33 Solution 2. Use arrows to indicate the graphs of each of the four inequalities. The solution set of the system is the intersection of those graphs. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 34 Solution 3. The life expectancies of U.S. females from 0 years through 20 years old in the calendar year 2015 are represented by the vertical line segment above t = 35 on the t-axis. We see the life expectancies are between 61.6 years and 81.1 years, inclusive. Copyright 2015, 2008, 2011 Pearson Education, Inc. Section 3.6, Slide 35 Solution We can also use the equations of B and T to find the life expectancies of U.S. females from 0 years through 20 years old in the calendar year Evaluate B and T at 35: B(35) = 0.097(35) 81.1 T(35) = 0.075(35) 61.6 This means the life expectancies are between 61.6 years and 81.1 years, inclusive, which checks.