Chapter 5 Linear Inequalities and Linear Programming Section 2 Systems of Linear Inequalities in Two Variables Solving Systems of Linear Inequalities Graphically We now consider systems of linear inequalities such as x+y>6 2x y > 0 We wish to solve such systems graphically, that is, to find the graph of all ordered pairs of real numbers (x, y) that simultaneously satisfy all the inequalities in the system.
The graph is called the solution region for the system (or feasible region.) To find the solution region, we graph each inequality in the system and then take the intersection of all the graphs. 2 Graphing a System of Linear Inequalities: Example To graph a system of linear inequalities such as 1 y x2 2
x 4 y we proceed as follows: Graph each inequality on the same axes. The solution is the set of points whose coordinates satisfy all the inequalities of the system. In other words, the solution is the intersection of the regions determined by each separate inequality. 3 Graph of Example The graph of the first inequality y < (1/2)x + 2 consists of the region
shaded yellow. It lies below the dotted line y = (1/2)x + 2. The graph of the second inequality is the blue shaded region is above the solid line x 4 = y. The graph is the region which is colored both blue and yellow. 4 Corner Points A corner point of a solution region is a point in the solution region that is the intersection of two boundary lines. In the previous example, the solution region had a
corner point of (4,0) because that was the intersection of the lines y = 1/2 x + 2 and y = x 4. Corner point 5 Bounded and Unbounded Solution Regions A solution region of a system of linear inequalities is bounded if it can be enclosed within a circle. If it cannot be enclosed within a circle, it is unbounded. The previous example had an unbounded solution region because it extended infinitely far to the left (and up and down.) We will
now see an example of a bounded solution region. 6 Graph of More Than Two Linear Inequalities To graph more than two linear inequalities, the same procedure is used. Graph each inequality separately. The graph of a system of linear inequalities is the area that is common to all graphs, or the intersection of the graphs of the individual inequalities. Example: 7
Application Suppose a manufacturer makes two types of skis: a trick ski and a slalom ski. Suppose each trick ski requires 8 hours of design work and 4 hours of finishing. Each slalom ski requires 8 hours of design and 12 hours of finishing. Furthermore, the total number of hours allocated for design work is 160, and the total available hours for finishing work is 180 hours. Finally, the number of trick skis produced must be less than or equal to 15. How many trick skis and how many slalom skis can be made under these conditions? How many possible answers? Construct a set of linear inequalities that can be used for this problem. 8
Application Solution x and y must Let x represent the number of both be positive trick skis and y represent the number of slalom skis. Then the Number of following system of linear trick skis has inequalities describes our to be less than problem mathematically.
or equal to 15 Actually, only whole numbers Constraint on for x and y should be used, but Constraint on the the total we will assume, for the moment number of number of that x and y can be any positive finishing hours design hours real number. 9 Application
Graph of Solution The origin satisfies all the inequalities, so for each of the lines we use the side that includes the origin. The intersection of all graphs is the yellow shaded region. The solution region is bounded and the corner points are (0,15), (7.5, 12.5), (15, 5), and (15, 0) 10
"Factoring"' 1. Common Factor . ab - ac + ad = a (b - c + d) *The greatest common factor is the largest number that can divide all the terms of the given expression.
Anxiety - the negative aspect of experiencing stress. State anxiety - anxiety felt in a specific situation. Trait anxiety - general levels of anxiety for a person in any situation. Somatic anxiety - the feelings of anxiety associated with the...
Databáze jsou všude kolem nás Úrovně abstrakce Fyzická úroveň popisuje, jak je uložen záznam (např. o zákazníkovi) Logická úroveň popisuje jaká data jsou v databázi uložena a vztahy mezi daty type customer = record customer_id: string; customer_name: string; customer_street: string;...
National Certificate National Diploma Technical Certificate 6 units 12 units 18 units = 1 'A' level = 2 'A' levels = 3 'A' levels Entry to Vocational Qualifications Level 3 Entry Criteria Direct Entry - 16 year olds 5 GCSEs...
All of the genes of a population of organisms. Organisms that are the most successful at reproducing contribute most to the gene pool. Evolution. Any change in the frequency of any allele within a gene pool. ... Population Evolution Last...
Recherche fondamentale alternative Difficile d'avoir le même niveau qu'une équipe de recherche. La théorie de Garret Lisi ne semble pas marcher : voir l'article de Marcus du Sautoy dans Telegraph 2008 : « E8 can be thought of as the...
Rote rehearsal - Repeating over and over again. Examples of Mnemonics that you've used? Popular ones that we might know? Forgetting. Forgetting is the act of not being able to recall information. In many cases this is a temporary loss....
Office of Information Technology IRS PROCUREMENT ORGANIZATION FACTS Obligate 1.8 Billion Per Year Spend 20% of IRS Budget 505 People Nationwide Manage 15.5 Billion in Contract Administration Contracting Support for Treasury in Several Major IT Awards Tips for Small Business...
Download Presentation
Ready to download the document? Go ahead and hit continue!