Hawkes Learning Systems: College Algebra

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”. • Applications of the term regions of constraint.

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 that 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 that half-plane solve the inequality. Otherwise, the points in the other half-plane solve the inequality. Shade in 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 1: Solving Linear Inequalities Solve the following linear inequality by graphing its solution set.

11. Example 2: Solving Linear Inequalities Solve the following linear inequality by graphing its solution set.

12. Example 3: 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 4: 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 5: 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 6: Inequalities Involving Absolute Values Graph the solution set in 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 6: Inequalities Involving Absolute Values (Cont.) We now intersect the solution sets to obtain the final answer:

19. Interlude: Regions of Constraint • One noteworthy application of the ideas in this section is linear programming, an important mathematical tool in business and the social sciences. • Linear programming is a method used to maximize or minimize a variable expression, subject to certain constraints on the values of the variables.

20. Interlude: Regions of Constraint • The first step in linear programming is to determine the region of constraint, a mathematical description of all the possible values that can be taken on by the variables (also called the feasible region). • If the variable expression to be maximized or minimized contains just two variables, the region of constraint will be a portion of the Cartesian plane, and it will usually be defined by the intersection of a number of half-planes.

21. Example 7: Regions of Constraint A family orchard is in the business of selling peaches and nectarines. The members of the family know that to prevent a certain pest infestation, the number of nectarine trees in their orchard cannot exceed the number of peach trees. Also, because of the space requirements of each type of tree, the number of nectarine trees plus twice the number of peach trees cannot exceed 100 trees. Graph the region of constraint for this situation. Let represent the number of peach trees and the number of nectarine trees. The following constraints are given:

22. Example 7: Regions of Constraint (Cont.) Graph of the intersection of the four half-planes that solve each individual inequality, with the horizontal axis representing and the vertical axis :