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Section 9.3 More Equations and Inequalities

Solving Linear and Compound Linear Inequalities in Two Variables 9.3 In Chapter 3, we learned how to solve linear inequalities in one variable. In this section, we will first learn how to graph the solution set of linear inequalities in two variables. Then we will learn how to graph the solution set of systems of linear inequalities in two variables. Define a Linear Inequality in Two Variables

Example 1 Solution The points that solve the inequality are points that are in the shaded region or on the line. Points that are not in the solution set are not part of the shaded region. Solutions in shaded region and on line. Solutions ofCheck by Substituting Into (-1,1) (0,0) (1,0) (on the line) Not in Solution setCheck by Substituting Into (2,0) FALSE (-2,0) FALSE (4,1) FALSE

Graph a Linear Inequality in Two Variables The line of the graph divides the plane into two regions or half planes The line x+y =1 is the boundary line between the two half planes. We can use this boundary and two different methods to graph a linear inequality in two variables. The first method we will discuss is the test point method.

Example 2 Solution 1)Graph the boundary line 3x + 2y = -6 as a solid line. It is a solid line since it has a less than or equal sign. 2) Choose a test point not on the line and substitute it into the inequality to determine whether or not it makes the inequality true. Since the test point (0,0) does not satisfy the inequality, we will shade the side of the line that does not contain the point (0,0).

Example 3 Solution 1)Graph the boundary line -x + 2y = -3 as a dotted line. It is a dotted line since it has a greater than sign. 2) Choose a test point not on the line and substitute it into the inequality to determine whether or not it makes the inequality true. Since the test point (0,0) does satisfy the inequality, we will shade the side of the line that does contain the point (0,0).

Another method we can use to graph the linear inequalities in two variables involves writing the boundary line in slope-intercept form.

Example 4 Solution Notice that the inequality is already in the correct form. If not in correct form rewrite.

Example 5 Solution Notice that the inequality is not in the correct form. First write in correct form.

Graph a Compound Linear Inequality in Two Variables Linear inequalities in two variables are called compound linear inequalities if they are connected by the words and or or. The solution set of a compound inequality containing and is the intersection of the solution sets of the inequalities. The solution set of a compound inequality containing or is the union of the solution sets of the inequalities.

Example 6 Solution

Any point in the solution set must satisfy both inequalities, and any point not in the solution set will not satisfy both inequalities. Check three test points one from each of the shaded regions. The solution set is shaded in blue. Since the inequalities contain the word and you Can also find the solution by selecting the intersection of both inequalities. That is Where both inequalities have both types of shading.

Example 7 Solution Graph each inequality separately. The solution set of the compound inequality will be the union (total) of the shaded regions. Everything that has been shaded will be part of the solution. You can use a check test point to verify the solution set is shaded in blue. The check test point must satisfy one or both inequalities.