Unit 5 Section 4

# Unit 5 Section 4 - 5.4 Systems of Inequalities Objectives...

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5.4 Systems of Inequalities Objectives: After completing this section, you should be able to: 1. Sketch the graphs of inequalities in two variables; 2. Solve systems of inequalities; 3. Use systems of inequalities in two variables to model and solve applied problems. In Unit 1, we discussed inequalities in one variable. In this section, we will expand our work with inequalities to include both a single inequality in two variables and a system of inequalities in two variables. Example 5.4.1 Examples of inequalities in Two Variables (a) 6 3 2 y x (b) 6 3 2 2 2 y x (c) 1 2 x y Graphing a Single Inequality in Two Variables Let’s start with a linear inequality . Actually, graphing a linear inequality is almost as easy as graphing a linear equation. But before we begin, we must discuss some important subsets of a plane in a rectangular coordinate system. A line divides a plane into two-halves called half-planes . A vertical line, say 2 x , divides a plane into left and right half-planes (Fig. 5.4.1 (a)); a nonvertical line, say x y 2 , divides a plane into upper and lower half-planes (Fig. 5.4.1 (b)). Keep This in Mind As with equations in two variables, a solution of an inequality in x and y is an ordered pair   b a , that makes the inequality true when a and b are substituted for x and y, respectively. A graph is often the most convenient way to represent solutions of inequalities in two variables. The graph of an inequality in two variables x and y consists of all points   y x , whose coordinates satisfy the inequality.

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Unit 5. Systems of Equations and Inequalities Section 4 page 2 _____________________________________________________________________________________ Consider again the line 1 x y . Let us investigate the half-planes determined by this linear equation. Note that four inequalities are formed from this equation by replacing the = sign by ≥, >, ≤, and < , respectively. (i) 1 x y (iii) 1 x y (ii) 1 x y (iv) 1 x y The graph of each is a half-plane . The line y = x -1 is called the boundary line or edge for the half- plane. Consider a point on the boundary line. If x = 2, we have y = 2 - 1 = 1. So the point (2, 1) is on the line y = x - 1. Note that points with x-coordinate 2 and y-coordinate less than 1 (e.g. (2, 0) and (2, -4)) are below the line. Thus, the half-plane below the line (or lower half-plane) corresponds to the solution of the inequality y < x - 1 . Similarly, the half-plane above the line (or upper half-plane) corresponds to the solution of the inequality y < x - 1 , as shown on Fig.5.4.2 at the left. The boundary line is included for ≥ and ≤ and excluded for > and <. 2
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Unit 5 Section 4 - 5.4 Systems of Inequalities Objectives...

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