# SOLVING TWO VARIABLE LINEAR INEQUALITIES INCLUDING ABSOLUTE VALUE INEQUALITIES

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• Slide 1
• SOLVING TWO VARIABLE LINEAR INEQUALITIES INCLUDING ABSOLUTE VALUE INEQUALITIES
• Slide 2
• Summary of Inequality Signs > < > < Continuous line Dashed line Shade above the line Shade below the line
• Slide 3
• Graphing Linear Inequalities The graph of a linear inequality is a region of the coordinate plane that is bounded by a line. This region represents the SOLUTION to the inequality.
• Slide 4
• A linear inequality is an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line.
• Slide 5
• Graph the following inequality: x > 2 Boundary is:x = 2 We shaded at the right of the line because x is more than 2. The line is dashed because it is not equal or less than x, so the line which is the boundary is not included in the solution.
• Slide 6
• Graph the following inequality: y < 6 Boundary: y = 0x + 6m= 0y- intercept = (0,6) We shaded below the line because y is less than 6. The line is dashed because it is not equal or less than y, so the line which is the boundary is not included in the solution.
• Slide 7
• Example
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• 1.The boundary line is dashed.
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• 2.Substitute (0, 0) into the inequality to decide where to shade. So the graph is shaded away from (0, 0).
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• Slide 11
• Graph the following inequality: 4x + 2y < 10 Solve for y y < -2x + 5 Boundary is: y = -2x + 5m= -2 y- intercept = (0,5) We shaded below the line because y is less than the expression -2x + 5. The line is dashed because it is not equal or less than y, so the line which is the boundary is not included.
• Slide 12
• Graph the following inequality: -9x + 3y< 3 Solve for y y < 3x + 1 Boundary is: y = 3x + 1 m= 3y- intercept = (0, 1) We shaded below the line because y is less than the expression 3x +1. The line is dashed because it is not equal or less than y, so the line which is the boundary is not included.
• Slide 13
• Graph the following inequality: y 2 > (x 4) Solve for y y > x 3 Boundary is: y = x 3 m = y-intercept = (0, -3) We shaded above because y is greater or equal than the expression and the line is continuous because the word equal in greater or equal indicates that the boundary is included in the solution.
• Slide 14
• EXAMPLE
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• Problem, cont
• Slide 19
• Graph the following absolute value equation: y = |x| For x 0 y = -x y = x Now lets shift it two units up: y = |x| + 2 Now lets shift it three units to the right: y = |x - 3| + 2 Now lets graph it upside down y = |x-3| + 2 Now lets make it skinner y = 6|x-3| + 2
• Slide 20
• So, thats how the different parameters in an absolute value equation affect our graph. Now lets graph absolute value inequalities.
• Slide 21
• Absolute Value Inequalities Graph the absolute value function then shade above OR below Solid liney Dashed liney Shade above y>, y> Shade belowy
• Absolute Value Inequalities Graph y + 1 < -2|x + 2| -y < -2|x + 2| - 1 y > 2|x + 2| + 1 -y so CHANGE the direction of the inequality
• Slide 28
• Absolute Value Inequalities y > 2|x + 2| + 1 Vertex = (-2, 1) Slope = 2 Solid line Shade above
• Slide 29
• Absolute Value Inequalities y > 2|x + 2| + 1
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• Absolute Value Inequalities y > 2|x + 2| + 1
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• Absolute Value Inequalities y > 2|x + 2| + 1
• Slide 32
• Absolute Value Inequalities y > 2|x + 2| + 1
• Slide 33
• Absolute Value Inequalities Write an equation for the graph below.
• Slide 34
• Graph the following inequality: y > |x| Finding the boundary: For x 0 y = -x y = x There are two regions: Testing point (0,2) 2 > | 0| 2 > 0 true Therefore, the region where (0,2) lies is the solution region and we shade it..
• Slide 35
• Finding the boundary: For x + 1 0 y = -(x+1) 3 y = x+1 3 y = - x 1 -3y = - x 4 y = x 2 There are two regions: Testing point (0,0) 0 < | 0+1| 3 0 < -2 false So the region where (0,0) lies is not in the solution region, therefore we shade the region below. Graph the following inequality: y < |x+1| 3.
• Slide 36
• Steps: 1.Decide if the boundary graph is solid or dashed. 2.Graph the absolute value function as the boundary. 3.Use the point (0, 0), if it is not on the boundary graph, to decide how to shade.
• Slide 37
• Graph y 2|x 3| + 2
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• Graph y 2|x 3| + 2 1.The boundary graph is solid.
• Slide 39
• 2.y 2|x 3| + 2 0 2|0 3| + 2 0 2|-3| + 2 0 6 + 2 0 8 False So shade away from (0, 0).
• Slide 40
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• 1.The boundary graph is dashed.
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• Your Turn! 8. Graph 9. Graph
• Slide 46
• 8. 9.
• Slide 47
• Example 10
• Slide 48
• y |x+4| - 3y 2x+5

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