What is a Linear Inequality in Two Variables?
A linear inequality in two variables is an inequality that can be written in one of the following forms, where A, B, and C are real numbers and A and B are not both zero:- (Ax + By < C)- (Ax + By ? C)- (Ax + By > C)- (Ax + By ? C)
These inequalities can be thought of as similar to linear equations but instead of defining a specific line on the coordinate plane, they define a region of the plane.
How do you Graph a Linear Inequality?
1. Convert the Inequality to an Equation: Change the inequality sign to an equals sign to find the boundary line. For example, the inequality (2x + 3y < 6) becomes the equation (2x + 3y = 6).
2. Graph the Boundary Line: - If the inequality is strict (< or >), draw a dashed line to indicate that the points on the line are not included in the solution. - If the inequality is non-strict (? or ?), draw a solid line to indicate that the points on the line are included in the solution.
3. Choose a Test Point: Select a point not on the boundary line to determine which side of the line should be shaded. The point (0,0) is a good choice unless it lies on the line.
4. Substitute the Test Point: Substitute the coordinates of the test point into the original inequality. - If the inequality is satisfied, shade the region that includes the test point. - If the inequality is not satisfied, shade the opposite side.
Example: Graphing a Linear Inequality
What are the steps to graph the inequality (3x - 2y ? 6)?
1. Convert to an Equation: Change (3x - 2y ? 6) to (3x - 2y = 6).
2. Graph the Boundary Line: - To graph (3x - 2y = 6), find the x-intercept and y-intercept: - x-intercept: Set y = 0, then (3x = 6) => (x = 2). - y-intercept: Set x = 0, then (-2y = 6) => (y = -3). - Plot the intercepts (2,0) and (0,-3) and draw a solid line through them because the inequality is ?.
3. Choose a Test Point: A convenient test point is (0,0).
4. Substitute the Test Point: Substitute (0,0) into (3x - 2y ? 6): [3(0) - 2(0) ? 6 => 0 ? 6 ] This is true.
5. Shade the Region: Since the test point (0,0) satisfies the inequality, shade the region that includes (0,0) and is below or on the boundary line.
By following these steps, you can effectively graph any linear inequality in two variables, ensuring you correctly represent the solution set on the coordinate plane.
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