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**6. 5 Graphing Linear Inequalities in Two Variables 7**

6.5 Graphing Linear Inequalities in Two Variables 7.6 Graphing Linear Inequalities System Objectives 1. Graph linear inequalities. 2. Graph systems of linear inequalities.

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Linear Inequalities A linear inequality in two variables is an inequality that can be written in the form Ax + By < C, where A, B, and C are real numbers and A and B are not both zero. The symbol < may be replaced with , >, or . The solution set of an inequality is the set of all ordered pairs that make it true. The graph of an inequality represents its solution set.

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**To Graph a Linear Inequality**

Step (1) Solve for y, convert the inequalities to Slope-Intercept Form. If one variable is missing, solve it and go to step (2). (2) Graph the related equation. ** If the inequality symbol is < or >, draw the line dashed. ** If the inequality symbol is or , draw the line solid. (3) If y < or y , shade the region BELOW the line If y > or y , shade the region ABOVE the line If x < or x , shade the region LEFT to the line If x > or x , shade the region RIGHT to the line

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**Example Graph y > x 4. Already in S-I form.**

The related equation is y = x 4. Use a dashed line because the inequality symbol is >. This indicates that the line itself is not in the solution set. (3) Determine which half-plane satisfies the inequality. y > x 4 “>” shade above

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**Example Graph: 4x + 2y 8 1. Convert to S-I form: y -2x + 4**

2. Graph the related equation. 3. Determine which half-plane satisfies the inequality. “y ” shade below

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**Example Graph 2x – 4 > 0 . 1. One variable is missing. Solve it:**

2. Graph the related equation x = 2 3. Determine which half-plane satisfies the inequality. “x >” shade the right.

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**Example Graph 8 - 3y 2 . 1. One variable is missing. Solve it: y 2**

2. Graph the related equation y = 2 3. Determine the region to be shaded. “y ” shaded below

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**7.6 Systems of Linear Inequalities**

Graph the solution set of the system. First, we graph x + y 3 using a solid line. y - x + 3 “above” Next, we graph x y > 1 using a dashed line. y < x – 1 “below” The solution set of the system of equations is the region shaded both red and green, including part of the line x + y = 3.

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Example Graph the following system of inequalities and find the coordinates of any vertices formed: We graph the related equations using solid lines. We shade the region common to all three solution sets.

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Example continued The system of equations from inequalities (1) and (3): y + 2 = 0 x + y = 0 The vertex is (2, 2). The system of equations from inequalities (2) and (3): x + y = 2 The vertex is (1, 1). To find the vertices, we solve three systems of equations. The system of equations from inequalities (1) and (2): y + 2 = 0 x + y = 2 The vertex is (4, 2).

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Summary To graph a two-variable linear inequality, graph the related equation first (variable y must be solved) with appropriate boundary line: “<” and “>” use dash line “≤” and “≥” use solid line “y < …” and “y ≤ …” shade the region below “y > …” and “y ≥ …” shade the region above “x < …” and “x ≤ …” shade the region left “x > …” and “x ≥ …” shade the region right When graphing the linear inequality system, follow the step 1 ~ 3 and choose the region shaded most.

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Assignment 6.5 P 363 #’s (even), (even), (even) 7.6 P 435 #’s

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