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**Linear Inequalities in Two Variables**

The graph of the solutions of a linear inequality in two variables is a half-plane. Procedure for Graphing Linear Inequalities 1. Rewrite the linear inequality as a linear equation. Change the inequality symbol ( < , > , , or ) to an equal sign. 2. The next step is to graph that line. If the inequality symbol has an equal sign ( or ), draw a solid line. If the inequality symbol does not have an equal sign (< , >), draw a dashed line. Pick any ordered pair that is not on the graphed line. This will be your test point. Substitute the coordinates of that ordered pair into the original inequality. If the inequality is true, shade the side of the line where the test point is. If the inequality is false, shade the other side of the line. Next Slide

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**x y 0 -4 Example 1. Graph: 3x +2y < 6 Solution:**

x axis y axis Example 1. Graph: 3x +2y < 6 Solution: Write the inequality as a linear equation, then graph. 3x + 2y = 6 Pick a test point not on the line, say (0,0). Since this is true, we can shade the side where the test point is. x y 0 -4 3 0 y axis x axis

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**Write the inequality as a linear equation, and graph.**

Example 2. Graph y > 2x Solution: x axis y axis Write the inequality as a linear equation, and graph. y = 2x Pick a test point, say (0,4). Since this is false, we must shade the side opposite the test point. Note: There is another method to determining which side of the line to shade. If the inequality is y> or y≥, shade above the line. If the inequality is y< or y≤, shade below the line. This example is y>2x, therefore we shade above the line. Using this method takes a little work to get y by itself but then you don’t have to worry about the test point. y axis x axis

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**Since this is false, shade on the opposite side of the test point.**

x axis y axis Since this is false, shade on the opposite side of the test point. Note: If the inequality is x> or x≥, shade to the right of the line. If the inequality is x< or x≤, shade to the left of the line. This example is x≤−4, therefore we shade to the left of the line. x axis y axis

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**Because our inequality is y >, shade above the dashed line.**

x axis y axis 1st, write the inequality as linear equation to graph the line. Either use the y intercept and the slope or choose values for x. If you choose values for x, always choose x=0 and also choose values that are multiples of the denominator in the fraction. Because our inequality is y >, shade above the dashed line. x axis y axis

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**The shaded area will be to the left.**

x axis y axis Recall that the word “and” indicates the intersection of the solutions sets of each inequality. The shaded area will be to the left. Answer The shaded area will be below the line. The intersection is the area that satisfied both inequalities. (shaded twice) x axis y axis

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**In the previous sections, we solved systems of equations such as:**

The solution set of the system is the intersection of the solution sets of the individual lines. (i.e., where the two lines meet.) The previous example contained the word “and”, which means the intersection. Another way of asking for the intersection is using a system such as: The solution set of the system is the intersection of the solution sets of the individual inequalities. (i.e., where the graph is shaded twice.) Next Slide

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**Example 6. Solve the following system by graphing.**

Solution: x axis y axis Next Slide Since y >, shade above. The solution is the area which was shaded twice. We will darken that area with another color to show the intersection more clearly. Since y ≥, shade above.

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**Solve the following system by graphing.**

Your Turn Problem #6. Solve the following system by graphing. x axis y axis Answer:

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**Example 7. Solve the following system by graphing.**

Solution: x axis y axis Therefore, our graph will only be shaded in Quadrant I, the upper right-hand quadrant. The solution is the area in Quadrant I which satisfies all four inequalities, shown here in the light blue color. Easier to graph last two inequalities using the intercept method. Then use test points to determine shaded area. Next Slide

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**Your Turn Problem #7. Graph: Answer: The End. B.R. 1-25-07 y axis**

x axis y axis (4,0) (0,4) (7,0) (0,6) Answer: The End. B.R.

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