1 / 18

Hawkes Learning Systems: College Algebra

Hawkes Learning Systems: College Algebra. 3.5: Linear Inequalities in Two Variables. Objectives. Solving linear inequalities in two variables. Solving linear inequalities joined by “and” or “or ”. Linear Inequalities in Two Variables.

oistin
Télécharger la présentation

Hawkes Learning Systems: College Algebra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Hawkes Learning Systems:College Algebra 3.5: Linear Inequalities in Two Variables

  2. Objectives • Solving linear inequalities in two variables. • Solving linear inequalities joined by “and” or “or”.

  3. Linear Inequalities in Two Variables If the equality symbol in a linear equation in two variables is replaced with or , the result is a linear inequality in two variables. A linear inequality in the two variables and is an inequality that can be written in the form Where , , and are constants and and are not both .

  4. Linear Inequalities in Two Variables • The solution set of a linear inequality in two variables consists of all the ordered pairs in the Cartesian plane that lie on one side of a line in the plane, possibly including those points on the line. • The first step in solving such an inequality is to identify and graph this line. • The line is simply the graph of the equation that results from replacing the inequality symbol in the original problem with the equality symbol.

  5. Linear Inequalities in Two Variables Any line in the Cartesian plane divides the plane into two half-planes, and, in the context of linear inequalities, all of the points in one of the two half-planes will solve the inequality. The green and pink portions of each graph are the half-planes; the blue lines are the boundary lines. This graph has an open half-plane (the line is not included in the solution set). This graph has a closed half-plane (the line is included in the solution set).

  6. Linear Inequalities in Two Variables The points on the line, called the boundary line in this context, will also solve the inequality if the inequality symbol is or , and this fact must be denoted graphically by using a solid line. The green and pink portions of each graph are the half-planes; the blue lines are the boundary lines. This graph has an open half-plane (the line is not included in the solution set). This graph has a closed half-plane(the line is included in the solution set).

  7. Solving Linear Inequalities in Two Variables Step 1: Graph the line in R2that results from replacing the inequality symbol with .

  8. Solving Linear Inequalities in Two Variables Step 2: Determine which of the half-planes solves the inequality by substituting a test point from one of the two half-planes into the inequality. If the resulting statement is true, all the points in the half-plane that contains the test-point solve the inequality. If the resulting statement is false, all the points in the other half-plane that did not contain the test-point solve the inequality. Shadein the half-plane that solves the inequality.

  9. Solving Linear Inequalities in Two Variables Select a Test Point Substitute into the Inequality true statement false statement Shade entire half-plane that includes the test point Shade entire half-plane that does not include the test point

  10. Example: Solving Linear Inequalities Solve the following linear inequality by graphing its solution set.

  11. Example: Solving Linear Inequalities Solve the following linear inequality by graphing its solution set.

  12. Example: Solving Linear Inequalities Solve the following linear inequality by graphing its solution set.

  13. Solving Linear Inequalities Joined by “And” or “Or” • In Section 1.2, we defined the union of two sets and , denoted , as the set containing all elements that are in set or set , and we defined the intersection of two sets and , denoted , as the set containing all elements that are in both and . • For the solution sets of two inequalities and , represents the solution set of the two inequalities joined by the word “or” and represents the solutions set of the two inequalities joined by the word “and”.

  14. Example: Linear Inequalities Joined by “And” or “Or” To find the solution sets in the following problems, we will solve each linear inequality individually and then form the union or the intersection of the individual solutions, as appropriate. Graph the solution set that satisfies the following inequalities.

  15. Example: Linear Inequalities Joined by “And” or “Or” Graph the solution set that satisfies the following inequality.

  16. Inequalities Involving Absolute Values • In Section 2.2, we saw that an inequality of the form can be rewritten as the compound inequality . • This can be rewritten as the joint condition and , so an absolute value inequality of this form corresponds to the intersection of two sets. • Similarly, an inequality of the form can be rewritten as , so the solution of this form of absolute value inequality is a union of two sets.

  17. Example: Inequalities Involving Absolute Values Graph the solution set in R2 that satisfies the joint conditions and . We need to identify all ordered pairs for which or while . That is, we need or while . The solution sets of the two conditions individually are:

  18. Example: Inequalities Involving Absolute Values (Cont.) We now intersect the solution sets to obtain the final answer:

More Related