Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 1 Chapter 3 Systems of Linear Equations

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide Using Linear Inequalities in One Variable to Make Predictions

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 3 Example: Using Models to Compare Two Quantities One Budget ® office rents pickup trucks for $39.95 per day plus $0.19 per mile. One U-Haul ® location charges $19.95 per day plus $0.49 per mile (Sources: Budget; U-Haul) 1. Find models that describe the one-day cost of renting a pickup truck from the companies. 2. Use graphs of your models to estimate for which mileages Budget offers the lower price.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 4 Solution 1. Let B(d) be the one-day cost (in dollars) of driving a Budget pickup truck d miles. Let U(d) be the one-day cost (in dollars) of driving a U-Haul pickup truck d miles. Equations of B and U are C = B(d) = 0.19d C = U(d) = 0.49d

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 5 Solution 2. Sketch a graph of B and U in the same coordinate system. Since the height of a point represents a price, Budget offers the lower price for mileages over approximately 66.7 miles.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 6 Addition Property of Inequalities If a < b, then a + c < b + c Similar properties hold for ≤, >, and ≥.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 7 Multiplication Property of Inequalities For a positive number c, if a < b, then ac < bc. For a negative number c, if a bc. Similar properties hold for ≤, >, and ≥. In words, when we multiply both sides of an inequality by a positive number, we keep the inequality symbol. When we multiply by a negative number, we reverse the inequality symbol.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 8 Linear inequality in one variable Definition A linear inequality in one variable is an inequality that can be put into one of the forms mx + b 0 mx + b ≤ 0mx + b ≥ 0 where m and b are constants and m ≠ 0.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 9 Solution of an inequality in one variable Definition We say a number is a solution of an inequality in one variable if it satisfies the inequality. The solution set of an inequality is the set of all solutions of the inequality. We solve an inequality by finding its solution set.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 10 Example: Solving a Linear Inequality Solve the inequality –2x ≥ 10.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 11 Solution Divide both sides of the inequality by –2, a negative number: Since we divided by a negative number, we reversed the direction of the inequality. The solution set is the set of all numbers less than or equal to –5.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 12 Reversing an Inequality Symbol Warning It is a common error to forget to reverse an inequality symbol when you multiply or divide both sides of an inequality by a negative number.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 13 Interval Notation We can use interval notation to describe the solution set of an inequality. An interval is the set of real numbers represented by the number line or by an unbroken portion of it. Examples of inequalities, their graphs, and interval notation are shown on the next slide.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 14 Words, inequalities, graphs, and interval notation

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 15 Example: Solving a Linear Inequality Solve –3(4x – 5) – 1 ≤ 17 – 6x. Describe the solution set as an inequality, in a graph, and in interval notation.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 16 Solution

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 17 Solution We can graph the solution set on a number line, or we can describe the solution set in interval notation as shown below:

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 18 Solution To verify, we check that for inputs greater than or equal to the outputs of y = –3(4x – 5) – 1 are less than or equal to the outputs of y = 17 – 6x. See the next slide for illustrations of the graphing calculator screens.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 19 Solution

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 20 Three-Part Inequalities 3 ≤ x ≤ 7 means the values of x are both greater than or equal to 3 and less than or equal to 7. We described the solution in a graph below and in interval notation by [3, 7].

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 21 Words, inequalities, graphs, and interval notations

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 22 Interval Notation vs. Ordered Pair Warning We use notation such as (3, 7) two ways: when we work with one variable, the interval (3, 7) is the set of numbers between 3 and 7; when we work with two variables, such as x and y, the ordered pair (3, 7) means x = 3 and y = 7.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 23 Example: Solving a Three-Part Inequality Solve –5 < 2x – 1 < 7.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 24 Solution Get x alone in the “middle part” of the inequality by applying the same operations to all three parts of the inequality:

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 25 Solution So, the solution set is the set of numbers between –2 and 4. We can graph the solution set on a number line, or we can describe the solution set in interval notation as (–2, 4).

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 26 Solution To verify our result, we check that, for values of x between –2 and 4, the graph of y = 2x – 1 is between the horizontal lines y = –5 and y = 7.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 27 Example: Using Models to Compare Two Quantities We can model the one-day pickup truck costs (in dollars) B(d) and U(d) at Budget and U-Haul, respectively, by the system C = B(d) = 0.19d C = U(d) = 0.49d where d is the number of miles driven. Use inequalities to estimate for which mileages Budget offers the lower price.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 28 Solution Budget offers the lower price when B(d) < U(d). Substitute 0.19d for B(d) and 0.49d for U(d) to get a linear inequality in one variable: 0.19d < 0.49d Solve the inequality by isolating d on the left side of the inequality (see the next slide):

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 29 Solution Budget offers the lower price if the truck is driven over miles.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.5, Slide 30 Solution To verify our result, we check that, for inputs greater than the outputs of y = 0.19x are less than the outputs of y = 0.49x