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Learn to graph inequalities in the coordinate plane and apply them in real-life scenarios. Understand open vs. closed half-planes for different inequalities, test points, and determine solutions.
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Objective • Graph inequalities in the coordinate plane.
Graph y > x + 1 The boundary line for this graph is the line y = x + 1. since the boundary is not part of the graph, it is shown as a dashed line on the graph. To determine which half-plane is the graph of y > x + 1, test a point NOT on the boundary. For example, you can test my favorite point (0, 0) the origin. Since 0 > 0 + 1 is false, (0, 0) is not a solution of y > x + 1. Thus, the graph is all points in the half-plane that does NOT contain (0, 0). This graph is called an open half-plane since the boundary is not part of the graph.
Graph y ≤ x + 1 The boundary line for this graph is also the line y = x + 1. Since the inequality y ≤ x + 1 means y < x + 1 or y = x + 1, this boundary is part of the graph. Therefore, the boundary is shown as a solid line on the graph. The origin (0, 0) is part of the graph of y ≤ x + 1 since 0 ≤ 0 + 1 is true. Thus, the graph is all points in the half plane that contains the origin and the line y = x + 1. This graph is called a closed half plane.
Graph y > -4x – 3 Graph the equation y = -4x – 3. Draw it as a dashed line since this boundary is NOT part of the graph. The origin, (0, 0), is part of the graph since 0 >-4(0) – 3 is true. Thus, the graph is all points in the half plane that contains the origin. CHECK: Test a point on the other side of the boundary, say (-2, -2). Since -2 > -4(-2) – 3 or -2 > 5 is false, (-2, -2) is NOT part of the graph.
Ex. 2: Graph 15x + 20y ≤ 240 to answer the application at the beginning of the lesson. How many of each ticket can Mr. Harris purchase? • First solve for y in terms of x. 15x + 20y ≤ 240 20y ≤ 240 – 15x y ≤ 12 - Then graph the equation as if it were equal to y as a solid line since the boundary is part of the graph. The origin (0, 0) is part of the graph since 15(0) + 20(0) ≤ 240 is true. Thus, the graph is all points in the half-plane that contains the origin.
Mr. Harris cannot buy fractional or negative numbers of tickets. • So, any point in the shaded region whose x- and y-coordinates are whole numbers is a possible solution. • For example, (5, 8) is a solution. This corresponds to buying five $15 tickets and eight $20 tickets for a total cost of 15(5) + 20(8) or $235.
Is the boundary included in the graphs of each inequality? 2x + y ≥ 3 Included 3x – 2y ≤ 1 Included 5x - 2 > 3y Not Included
Included or not? • Signs with < or > are NOT included • Signs with ≤ or ≥ are included • If the graph is < or >, then it is said to be an open half plane since the boundary is not part of the graph. • If a graph is ≤ or ≥, then it is a closed half-plane because the boundary is part of the graph. • Greater than → shade above • Less than → shade below
Determine which (-2,2), (4,-1), or (3,1) are solutions to the inequality x + 2y ≥ 3 x + 2y ≥ 3 -2 + 2(2) ≥ 3 -2 + 4 ≥ 3 2 ≥ 3 x + 2y ≥ 3 4 + 2(-1) ≥ 3 4 – 2 ≥ 3 2 ≥ 3 x + 2y ≥ 3 3 + 2(1) ≥ 3 3 + 2 ≥ 3 5 ≥ 3 Yes, this is the one
Graph each inequality g(x)>0 and f(x)≥(⅓)x The overlap is where these two meet which is where your answer to both of these lies.
