Linear inequalities in two variables


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Linear inequalities in two variables

  1. 1. Linear Inequalities inTwo Variables
  2. 2. I. Linear Inequalities in Two Variables A. Cartesian Coordinate System B. Points on the Cartesian Coordinate PlaneII. Special Product A. Multiplication by a Monomial B. Sum and Difference of Two Binomials C. Products of Two Binomials with like Terms D. Square of a BinomialIII. Factoring A. Common Factor B. Difference of Two Squares C. Perfect Square Trinomial D. Factoring by Completing the Square E. Quadratic Trinomial
  3. 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. 4. Linear Inequalities in Two Variables 3x + 2y > 4 -x + 3y < 2 -x + 4y ≥ 3 4x – y ≤ 4
  5. 5. The solutions of a linear inequality intwo variables x and y are the orderedpairs of numbers (x, y) that satisfythe inequality.Given an inequality: 4x – 7 ≤ 4 check if the following points are solutions to thegiven inequality. a.(0, -1) b.(2, 3) c.(-1, 1)
  6. 6. Since there will be an infinite numberof points in the solution, this is bestrepresented by a graph.
  7. 7. Cartesian Coordinate SystemA coordinate system in which the coordinates ofa point are its distances from a set ofperpendicular lines that intersect at an origin,such as two lines in a plane or three in space.
  8. 8. Rene Descartes – the one who invented theCartesian Coordinate System• It provided the first systematic link betweenEuclidean geometry and algebra.• Cartesian coordinates are the foundation of analyticgeometry, and provide enlightening geometricinterpretations for many other branches ofmathematics, such as linear algebra, complexanalysis, differentialgeometry, multivariate calculus, group theory, andmore.
  9. 9. • The development of the Cartesian coordinatesystem would play an intrinsic role in thedevelopment of the calculus by IsaacNewton and Gottfried Wilhelm Leibniz.• Many other coordinate systems have beendeveloped since Descartes, such as the polarcoordinates for the plane, andthe spherical and cylindrical coordinates for three-dimensional space.
  10. 10. How to Graph a Linear Inequality1. 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. 11. Assignment:Bring the following on Monday. • graphing papers • coloring materials • ruler