Linear Inequalities



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Chapter 3 : Linear Inequalities



Linear Inequalities arrow_upward


  • A linear inequality involves a linear expression by using any of the relational symbols such as <, >, ≤ or ≥.

  • Example:

  • x + 1 < 5
  • Note that the pointed arrow is always pointing to smaller number.
  • Inequalities can have one of the following symbols:
    • < Stands for less than.
    • > Stands for greater than.
    • ≤ Stands for less than or equal to.
    • ≥ Stands for greater than or equal to.
  • The < and > signs defines the sense of the inequality.
  • Two inequalities are said to have the same sense if the signs of inequality point in the same direction.

  • Example:

  • x + 3 > 2 and x + 1 > 0
  • Two inequalities are said to have the opposite sense if the signs of an inequality point in the opposite direction.

  • Example:

  • x - 4 < 0 and x > - 4

  • Graphical Representation of Inequalities arrow_upward


  • When we plot an inequality on a number line we can use an open hole or closed hole at that point.
  • An open hole means that the point is not included.
  • A closed hole means that the point is included.
  • Graph of X > 10
    • Note that the circle or hole at X = 10 is open, the arrow is pointing to number greater than 10.
  • Graph of X ≥ 10
    • Note that the circle or hole at X = 10 is closed, the arrow is pointing to a number greater than or equal to 10.
  • Graph of -3 > X or X < -3
    • Note that the circle or hole at X = -3 is open, the arrow is pointing to a number less than -3.
  • Graph of – 3 ≥ X or X ≤ -3
    • Note that the circle or hole at X = -3 is closed and arrow is pointing to numbers less than or equal to -3.

    Properties Used to Solve Linear Inequalities arrow_upward



    Addition Property:

  • If a > b then
    • a + c > b + c

    Subtraction Property:

  • If a > b then
    • a – c > b – c 

    Multiplication Property:  when c is positive:

  • If a > b and c > 0 then
    •  a × c > b × c 

    Multiplication Property:  when c is negative:

  • If a > b and c < 0 then
    • a × c < b × c

    Division Property: when c is positive:

  • If a > b and c > 0 then
    • a/c > b/c

    Division Property: when c is negative:

  • If a > b and c < 0 then
    • a/c < b/c

    System of Linear Inequalities arrow_upward


  • A system of linear equations consists of two or more linear inequalities:
    • 2x + 2y > 12
    • 2x - 2y > 0
  • The solution of a system of linear inequalities in two variables is any ordered pair that satisfies both of the linear inequalities.
  • We can find the solution graphically:
    • On plotting the ordered pairs on graph that satisfy the inequalities in the system.
  • The graph is called the solution region for the system.

  • Graphing Inequalities in Two Variables arrow_upward


  • Graph showing x – y < 2
  • Find the ordered pairs that satisfy the inequality.
  • Replace the inequality symbol with an equal sign and graph the corresponding linear equation.
    • If inequality is ≤ or ≥:  use solid line
    • If inequality is < or >: use dashed line
  • Choose an assumed point in one of the half-planes that is not on the line.
    • Substitute the coordinates of the test point into the inequality.
    • If it makes the statement true, shade the half-plane containing this test point.
    • If it makes the statement false, shade the half-plane not containing this test point.

    Solving Inequalities in Two Variables arrow_upward



    Example:

  • Solve the two variable inequality for 2y + 3x > 5.

  • Solution:

  • Given: 2y + 3x > 5
  • Write the given equation in a slope intercept form:
  •     2y + 3x = 5

  • Subtract 3x on both sides:
  •     2y + 3x – 3x = 5 – 3x

        2y = 5 – 3x

  • Now, divide 2 on both sides:
  •     2y/2 = -3x/2 + 5/2

        y = -1.5x + 2.5

  • To find the y-intercept, put x = 0,
  •     y = 0 + 2.5

        y = 2.5

  • The y-intercept is 2.5
  • To find x-intercept, put y = 0, we get:
  •     0 = -1.5x + 2.5

  • Subtract 2.5 on both sides:
  •      0 - 2.5 = -1.5x + 2.5 - 2.5

        -2.5 = -1.5x

  • Divide 1.5 on both sides:
  •     -2.5/1.5 = -1.5x/1.5

         -1.67 = -x

          x = 1.67        

  • The x-intercept is 1.67.


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