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Introduction Solving a linear inequality in two variables is similar to graphing a linear equation, with a few extra steps that will be explained on the following slides. Remember that inequalities have infinitely many solutions and all the solutions need to be represented. This will be done through the use of shading. 1 2.3.1: Solving Linear Inequalities in Two Variables

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Key Concepts A linear inequality in two variables has a half plane as the set of solutions. A half plane is a region containing all points that has one boundary, which is a straight line that continues in both directions infinitely. 2 2.3.1: Solving Linear Inequalities in Two Variables

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Key Concepts, continued To determine the solution set, first graph the inequality as a line. Sometimes the line or the boundary is part of the solution; this means it’s inclusive. Inequalities that have “greater than or equal to” (≥) or “less than or equal to” (≤) symbols are inclusive. Use a solid line when graphing the solution to inclusive inequalities. 3 2.3.1: Solving Linear Inequalities in Two Variables

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Key Concepts, continued Other times the line or boundary is NOT part of the solution; in other words, it’s non-inclusive. Inequalities that have “greater than” (>) or “less than” (<) symbols are non-inclusive. Use a dashed line when graphing the solution to non- inclusive inequalities. Either all the points above the line or all the points below the line will be part of the solution. To find out which side of the line contains the solutions, choose a point that is clearly on one side of the line or the other. 4 2.3.1: Solving Linear Inequalities in Two Variables

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Key Concepts, continued Substitute the test point into the inequality. If the test point makes the inequality true, shade the side of the line (half plane) that contains the test point. If it does not make the inequality true, shade the opposite side of the line. Shading indicates that all points in that region are solutions. 5 2.3.1: Solving Linear Inequalities in Two Variables

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Key Concepts, continued Graphing Equations Using a TI-83/84: 2.3.1: Solving Linear Inequalities in Two Variables 6 Step 1: Press [Y=] and arrow over to the left two times so that the cursor is blinking on the “\”. Step 2: Press [ENTER] two times for the greater than icon “ ” and three times for the less than icon “ ”. Step 3: Arrow over to the right two times so that the cursor is blinking after the equal sign. Step 4: Key in the equation using [X, T, Θ, n] for x. Step 5: Press [WINDOW] to change the viewing window, if necessary. Step 6: Enter in appropriate values for Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl, using the arrow keys to navigate. Step 7: Press [GRAPH].

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Key Concepts, continued Graphing Equations Using a TI-Nspire: 2.3.1: Solving Linear Inequalities in Two Variables 7 Step 1: Press the home key. Step 2: Arrow over to the graphing icon (the picture of the parabola or the U-shaped curve) and press [enter]. Step 3: At the blinking cursor at the bottom of the screen, press once the backspace key (a left facing arrow). A menu pops up that gives choices for less than or equal to (≤), less than ( ), and greater than or equal to (≥). Choose the appropriate symbol by using the arrow keys to navigate to the desired symbol and press the center button of the navigation pad. Alternatively, enter the number that is associated with the symbol. Step 4: Enter in the equation and press [enter].

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Key Concepts, continued 2.3.1: Solving Linear Inequalities in Two Variables 8 Step 5: To change the viewing window: press [menu], arrow down to number 4: Window/Zoom, and click the center button of the navigation pad. Step 6: Choose 1: Window settings by pressing the center button. Step 7: Enter in the appropriate XMin, XMax, YMin, and YMax fields. Step 8: Leave the XScale and YScale set to auto. Step 9: Use [tab] to navigate among the fields. Step 10: Press [tab] to “OK” when done and press [enter].

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Key Concepts, continued 9 Graphing a Linear Inequality in Two Variables 1.Determine the symbolic representation (write the inequality using symbols) of the scenario if given a context. 2.Graph the inequality as a linear equation. 3.If the inequality is inclusive (≤ or ≥), use a solid line. 4.If the inequality is non-inclusive ( ), use a dashed line. 5.Pick a test point above or below the line. 6.If the test point makes the inequality true, shade the half plane that contains the test point. 7.If the test point makes the inequality false, shade the half plane that does NOT contain the test point. 2.3.1: Solving Linear Inequalities in Two Variables

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Key Concepts, continued Standard Form of Linear Equations and Inequalities Linear equations can also be written as ax + by = c, where a, b, and c are real numbers. Similarly, an inequality can be written in the same form but with an inequality symbol (, ≤, or ≥) instead of an equal sign. To convert to slope-intercept form (y = mx + b), solve the equation or inequality for y. Remember to switch the inequality symbol if you multiply or divide by a negative. 10 2.3.1: Solving Linear Inequalities in Two Variables

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Key Concepts, continued Intercepts An intercept is the point at which the line intersects (or intercepts) the x- or y-axis. You have dealt with the y-intercept, which is the point at which the line intersects the y-axis. When an equation is in slope-intercept form, y = mx + b, b is the y-intercept. The general coordinates for the y-intercept are (0, y). Notice the x-coordinate of the y-intercept is 0. To solve for the x-intercept in an equation, set x = 0 and solve for y. 11 2.3.1: Solving Linear Inequalities in Two Variables

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Key Concepts, continued The x-intercept is the point at which the line intersects the x-axis. The general coordinates for the x-intercept are (x, 0). To solve for the x-intercept in an equation, set y = 0 and solve for x. You can plot a line using the intercepts. Find the x- and y-intercepts and then connect the points. Plotting a line using the intercepts is helpful in linear inequalities that are in context. Generally, linear inequalities in context have the constraint that the variables can’t be negative. This means the line will stop at the intercepts. 12 2.3.1: Solving Linear Inequalities in Two Variables

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Common Errors/Misconceptions using a solid line for a non-inclusive inequality or a dashed line for an inclusive inequality shading the region that makes the inequality false forgetting to switch the inequality symbol when multiplying or dividing by a negative if converting to slope-intercept form forgetting to shade the half plane 13 2.3.1: Solving Linear Inequalities in Two Variables

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Guided Practice Example 1 Graph the solutions to the following inequality. y > x + 3 14 2.3.1: Solving Linear Inequalities in Two Variables

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Guided Practice: Example 1, continued 1.Graph the inequality as a linear equation. Since the inequality is non-inclusive, use a dashed line. y = x + 3 To graph the line, plot the y-intercept first, (0, 3). Then use the slope to find a second point. The slope is 1. Count up one unit and to the right one unit and plot a second point. Connect the two points and extend the line to the edges of the coordinate plane. 15 2.3.1: Solving Linear Inequalities in Two Variables

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Guided Practice: Example 1, continued 2.Pick a test point above or below the line and substitute the point into the inequality. Choose (0, 0) because this point is easy to substitute into the inequality. y > x + 3 (0) > (0) + 3 0 > 3 This is false! 17 2.3.1: Solving Linear Inequalities in Two Variables

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Guided Practice: Example 1, continued 3.Shade the appropriate half plane. Since the test point makes the inequality false, all points on that side of the line make the inequality false. Shade above the line instead; this is the half plane that does NOT contain the point. 2.3.1: Solving Linear Inequalities in Two Variables 19

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2.3.1: Solving Linear Inequalities in Two Variables 20 ✔

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Guided Practice: Example 1, continued 21 2.3.1: Solving Linear Inequalities in Two Variables

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Guided Practice Example 3 A company that manufactures MP3 players needs to hire more workers to keep up with an increase in orders. Some workers will be assembling the players and others will be packaging them. The company can hire no more than 15 new employees. Write and graph an inequality that represents the number of new workers who can be hired. 22 2.3.1: Solving Linear Inequalities in Two Variables

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Guided Practice: Example 3, continued 1.Create an inequality from the context. There are two jobs to perform. Let x = the number of workers who will assemble the MP3 players. Let y = the number of workers who will package the MP3 players. x + y ≤ 15 23 2.3.1: Solving Linear Inequalities in Two Variables

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Guided Practice: Example 3, continued 2.Graph the inequality as a linear equation. Since the inequality is inclusive, use a solid line. x + y = 15 To graph the line, convert the standard form of the equation to slope-intercept form. y = –x + 15 Plot the y-intercept first, (0, 15). Then use the slope to find a second point. The slope is –1. Count down one unit and to the right one unit and plot a second point. Connect the two points and extend the line. 24 2.3.1: Solving Linear Inequalities in Two Variables

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Guided Practice: Example 3, continued 3.Determine where to stop the line. Stop the line at the intercepts because there cannot be negative employees. To find the y-intercept, look at the equation in slope- intercept form. The y-intercept is 15. The y-intercept coordinates are (0, 15). To find the x-intercept, use the standard form of the equation and set y = 0. x + y = 15 x + (0) =15 x = 15 25 2.3.1: Solving Linear Inequalities in Two Variables

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Guided Practice: Example 3, continued The coordinates of the x-intercept are (15, 0). 2.3.1: Solving Linear Inequalities in Two Variables 26

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Guided Practice: Example 3, continued 4.Pick a test point above or below the line and substitute the point into the inequality. Choose (0, 0) because this point is easy to substitute into the inequality. x + y ≤ 15 (0) + (0) ≤ 15 0 ≤ 15 This is true! 2.3.1: Solving Linear Inequalities in Two Variables 27

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2.3.1: Solving Linear Inequalities in Two Variables 28 Guided Practice: Example 3, continued

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2.3.1: Solving Linear Inequalities in Two Variables 29 Guided Practice: Example 3, continued 5.Shade the appropriate half plane. Since the test point makes the inequality true, that means all points on that side of the line make the inequality true. Shade the half plane that contains the test point.

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2.3.1: Solving Linear Inequalities in Two Variables 30 Guided Practice: Example 3, continued

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2.3.1: Solving Linear Inequalities in Two Variables 31 Guided Practice: Example 3, continued 6.Reduce the shading to fit the context of the problem and add labels. Having negative employees doesn’t make sense. Stop the shading at the x-axis, the y-axis, and the boundary line so that the shading ends at (0, 15) and (15, 0).

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2.3.1: Solving Linear Inequalities in Two Variables 32 Guided Practice: Example 3, continued x + y ≤ 15 ✔

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Guided Practice: Example 3, continued 33 2.3.1: Solving Linear Inequalities in Two Variables

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Graphing Linear Inequalities in Two Variables

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