Linear inequalities in two variables
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The document discusses linear inequalities in two variables and their graphical representations. It introduces the Cartesian coordinate system developed by Rene Descartes and its importance. It explains how to graph linear inequalities by first drawing the line as an equation, then determining whether to shade above or below the line based on whether a test point satisfies the inequality. Students are assigned to bring graphing paper, coloring materials, and a ruler to class on Monday to graph linear inequalities.
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- 3. IV. Statistics A. Introduction 1. Historical Background 2. Importance of Summation Notation B. Summation Notation C. Frequency Table 1. Definition of Terms 2. Frequency Distribution Table
- 4. Linear Inequalities in Two Variables 3x + 2y > 4 -x + 3y < 2 -x + 4y ≥ 3 4x – y ≤ 4
- 5. The solutions of a linear inequality in two variables x and y are the ordered pairs of numbers (x, y) that satisfy the inequality. Given an inequality: 4x – 7 ≤ 4 check if the following points are solutions to the given inequality. a.(0, -1) b.(2, 3) c.(-1, 1)
- 6. Since there will be an infinite number of points in the solution, this is best represented by a graph.
- 7. Cartesian Coordinate System A coordinate system in which the coordinates of a point are its distances from a set of perpendicular lines that intersect at an origin, such as two lines in a plane or three in space.
- 8. Rene Descartes – the one who invented the Cartesian Coordinate System • It provided the first systematic link between Euclidean geometry and algebra. • Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.
- 9. • The development of the Cartesian coordinate system would play an intrinsic role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz. • Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three- dimensional space.
- 10. How to Graph a Linear Inequality 1. Replace the inequality symbol with an equal sign. 2. Draw the graph of the equation in step 1. If the original inequality contains the symbol > or <, draw the graph using a dashed line. If the original inequality contains the symbol ≥ or ≤, draw the graph using a solid line. 3. Choose an arbitrary test point not on the line. The point (0, 0) is often convenient to use. Substitute this test point into the inequality. 4. (a) If the test point satisfies the inequality, shade the region on the side of the line containing this point. (b) If the test point does not satisfy the inequality, shade the region on the side of the line not containing this point.
- 11. Assignment: Bring the following on Monday. • graphing papers • coloring materials • ruler