Hawkes Learning Systems: College Algebra

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Copyright © 2011 Hawkes Learning Systems. All rights reserved.

Hawkes Learning Systems:College Algebra

3.5: Linear Inequalities in Two Variables




Objectives

o Solving linear inequalities in two variables.

o Solving linear inequalities joined by “and” or “or”.




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 .

, , ,

, , , ,ax by c ax by c ax by c ax by c

x y

a b c a b0





o 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.

o The first step in solving such an inequality is to identify and graph this line.

o The line is simply the graph of the equation that results from replacing the inequality symbol in the original problem with the equality symbol.





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.

This graph has a closed half-plane (the line is included in the solution set).

This graph has an open half-plane (the line is not included in the solution set).

The green and pink portions of each graph are the half-planes; the blue lines are the boundary lines.





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.

This graph has a closed half-plane (the line is included in the solution set).

This graph has an open half-plane (the line is not included in the solution set).

The green and pink portions of each graph are the half-planes; the blue lines are the boundary lines.




Solving Linear Inequalities in Two Variables

Step 1: Graph the line in R2 that results from replacing the inequality symbol with .

Solid Line Dashed Line

Non-strict. Strict.

Points on the line included in the solution set.

Points on the line excluded from the solution set.

or or





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. Shade in the half-plane that solves the inequality.





Select a Test Point

Substitute into the Inequality

Shade entire half-plane that includes the test

point

Shade entire half-plane that does not include

the test point

true statement false statement




Example: Solving Linear Inequalities

Solve the following linear inequality by graphing its solution set. 3 2 12x y

-intercept: 4,0x

-intercept: 0,6y





Solve the following linear inequality by graphing its solution set.

0x y





Solve the following linear inequality by graphing its solution set.

3x




Solving Linear Inequalities Joined by “And” or “Or”

o 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 .

o 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”.

AB A B

A BA B A B

AB

A B A B

A B




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.

5 2 10 and x y y x y x

5 2 10x y




Example: Linear Inequalities Joined by “And” or “Or”

Graph the solution set that satisfies the following inequality. 4 or 4x y x




Inequalities Involving Absolute Values

o In Section 2.2, we saw that an inequality of the form can be rewritten as the compound inequality .

o This can be rewritten as the joint condition and , so an absolute value inequality of this form corresponds to the intersection of two sets.

o 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.

x aa x a

x a x a

x a or x a x a



Copyright © 2011 Hawkes Learning Systems. All rights reserved.Example: Inequalities Involving Absolute

ValuesGraph the solution set in R2 that satisfies the joint conditions and . 3 1x 2 3y

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:

3 1 x 3 1x 3 2 3y 2x 4x 1 5y

2 or 4x x 1 5y



Copyright © 2011 Hawkes Learning Systems. All rights reserved.Example: Inequalities Involving Absolute

Values (Cont.)We now intersect the solution sets to obtain the final answer:

3 1 and 2 3x y

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Hawkes Learning Systems: College Algebra