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 1

Chapter 3Systems of Linear Equations


3.5 Using Linear Inequalities in One Variable to Make

Predictions


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.


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 + 39.95C = U(d) = 0.49d + 19.95


Solution2. 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.


Addition Property of Inequalities

If a < b, then a + c < b + c

Similar properties hold for ≤, >, and ≥.


Multiplication Property of Inequalities

• For a positive number c, if a < b, then ac < bc.• For a negative number c, if a < b, then ac > 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.


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 mx + b ≥ 0

where m and b are constants and m ≠ 0.


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.


Example: Solving a Linear Inequality

Solve the inequality –2x ≥ 10.


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.

2 10x 2 102 2

xx

5x


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.


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.


Words, inequalities, graphs, and interval notation


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.


Solution 3 4 5 1 17 6x x 12 15 1 17 6x x

12 14 17 6x x 12 14 17 66 6xx x x

6 14 17x 146 14 1 147x 6 3x 6 36 6x

12

x


Solution

We can graph the solution set on a number line, or we can describe the solution set in interval notation as shown below:


Solution

To verify, we check that for inputs greater than or1

,2

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.


Solution


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].


Words, inequalities, graphs, and interval notations


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.


Example: Solving a Three-Part Inequality

Solve –5 < 2x – 1 < 7.


Solution

Get x alone in the “middle part” of the inequality by applying the same operations to all three parts of the inequality: 5 2 1 7x

5 21 1711x

4 2 8x

24

2 22 8x

2 4x


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).


SolutionTo 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.


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 + 39.95C = U(d) = 0.49d + 19.95

where d is the number of miles driven. Use inequalities to estimate for which mileages Budget offers the lower price.


Solution

Budget offers the lower price when B(d) < U(d). Substitute 0.19d + 39.95 for B(d) and 0.49d + 19.95 for U(d) to get a linear inequality in one variable:

0.19d + 39.95 < 0.49d + 19.95

Solve the inequality by isolating d on the left side of the inequality (see the next slide):


Solution0.19 39.95 0.49 19.95d d

0.490.19 39.95 0.49 19.95 0.49d dd d 0.30 39.95 19.95d

39.950.30 39.95 19.9 39.955d 0.30 20d 0.30 200.30 0.30

d

66.6d

Budget offers the lower price if the truck is driven over miles.66.6


Solution

To verify our result, we check that, for inputs greater than the outputs of y = 0.19x + 39.95 are less than the outputs of y = 0.49x + 19.95.

66.6,

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