Systems of Inequalities

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Section 5.7

• Systems of Inequalities
• And Linear Programming

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Linear Inequality in Two Variables

• An inequality that can be written as
• Ax By lt C or Ax By gt C,
• where A, B, and C are real numbers
• and A and B are not both 0.
• The symbol lt may be replaced with ?, gt, or ?.

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Linear Inequalities

• The solution set of an inequality is the set of
  all ordered pairs that make it true.
• The graph of an inequality represents its
  solution set.

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Graphing an Inequality

• Draw the boundary line
• Make the inequality an equation.
• Graph the equation.
• gt or lt Solid line
• gt or lt Dashed line
• Choose a test point. (Any point not on the
  graph.)
• Substitute test point into original inequality.
• Shade the appropriate region.
• Shade the region that includes the test point if
  it makes the inequality true.
• If the test point does not make the inequality
  true, shade the other side of the line.

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Example

• Graph y gt x ? 4.
• We begin by graphing the related equation
  y x ? 4.
• We use a dashed line because the inequality
  symbol is gt. This indicates that the line itself
  is not in the solution set.

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Example continued

• Determine which half-plane satisfies the
  inequality by choosing a test point.

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To Graph a Linear InequalityA Recap

• Replace the inequality symbol with an equals sign
  and graph this related equation. If the
  inequality symbol is lt or gt, draw the line
  dashed. If the inequality symbol is ? or ?, draw
  the line solid.
• The graph consists of a half-plane on one side of
  the line and, if the line is solid, the line as
  well. To determine which half-plane to shade,
  test a point not on the line in the original
  inequality. If that point is a solution, shade
  the half-plane containing that point. If not,
  shade the opposite half-plane.

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Example

• Graph 4x 2y ? 8
• Graph the related equation, using a solid line.

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Example continued

• Determine which half-plane to shade by choosing
  a test point.

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Example

• Graph x gt 2 on a plane.
• 1. Graph the related equation.
• 2. Pick a test point (0, 0).
• x gt 2
• 0 gt 2 False
• Because (0, 0) is not a solution, we shade the
  half-plane that does not contain that point.

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Example

• Graph y ? 2 on a plane.
• 1. Graph the related equation.
• 2. Select a test point (0, 0).
• y ? 2
• 0 ? 2 True
• Because (0, 0) is a solution, we shade the
  region containing that point.

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Systems of Linear Inequalities

• Graph the solution set of the system.
• First, we graph x y ? 3 using a solid line.
• Choose a test point (0, 0) and shade the
  correct plane.
• Next, we graph x ? y gt 1 using a dashed line.
• Choose a test point and shade the correct plane.

The solution set of the system of equations is
the region shaded both red and green, including
part of the line x y ? 3.
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Example

• Graph the following system of inequalities and
  find the coordinates of any vertices formed

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Example continued
We graph the related equations using solid lines.
We shade the region common to all three solution
sets.
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Example continued
To find the vertices, we solve three systems of
equations.

• The system of equations from inequalities (1) and
  (2)
• y 2 0
• ?x y 2
• The vertex is (?4, ?2).
• The system of equations from inequalities (1) and
  (3)
• y 2 0
• x y 0
• The vertex is (2, ?2).
• The system of equations from inequalities (2) and
  (3)
• ?x y 2
• x y 0
• The vertex is (?1, 1).

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Linear Programming

• In many applications, we want to find a maximum
  or minimum value. Linear programming can tell us
  how to do this.
• Constraints are expressed as inequalities. The
  solution set of the system of inequalities made
  up of the constraints contains all the feasible
  solutions of a linear programming problem.
• The function that we want to maximize or minimize
  is called the objective function.

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Linear Programming Procedure

• To find the maximum or minimum value of a linear
  objective function subject to a set of
  constraints
• 1. Set up objective function and define
  constraints.
• 2. Graph the region of feasible solutions.
• 3. Determine the coordinates of the vertices of
  the region.
• 4. Evaluate the objective function at each
  vertex. The largest and smallest of those values
  are the maximum and minimum values of the
  function, respectively.

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Example

• A tray of corn muffins requires 4 cups of milk
  and 3 cups of wheat flour.
• A tray of pumpkin muffins requires 2 cups of
  milk and 3 cups of wheat flour.
• There are 16 cups of milk and 15 cups of wheat
  flour available, and the baker makes 3 per tray
  profit on corn muffins and 2 per tray profit on
  pumpkin muffins.
• How many trays of each should the baker make in
  order to maximize profits?

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Example continued

• A tray of corn muffins requires 4 cups of milk
  and 3 cups of wheat flour.
• A tray of pumpkin muffins requires 2 cups of
  milk and 3 cups of wheat flour.
• There are 16 cups of milk and 15 cups of wheat
  flour available, and the baker makes 3 per tray
  profit on corn muffins and 2 per tray profit on
  pumpkin muffins.
• Solution
• We let x the number of corn muffins and
  y the number of pumpkin muffins.
• Then the profit P is given by the function
  P 3x 2y.

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Example continued

• A tray of corn muffins requires 4 cups of milk
  and 3 cups of wheat flour.
• A tray of pumpkin muffins requires 2 cups of
  milk and 3 cups of wheat flour.
• There are 16 cups of milk and 15 cups of wheat
  flour available, and the baker makes 3 per tray
  profit on corn muffins and 2 per tray profit on
  pumpkin muffins.
• We know that x muffins require 4 cups of milk and
  y muffins require 2 cups of milk. Since there are
  no more than 16 cups of milk, we have one
  constraint. 4x 2y ? 16
• Similarly, the muffins require 3 and 3 cups of
  wheat flour. There are no more than 15 cups of
  flour available, so we have a second constraint.
  
• 3x 3y ? 15
• We also know x ? 0 and y ? 0 because the baker
  cannot make a negative number of either muffin.

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Example continued

• Thus we want to maximize the objective function
  P 3x 2y
• subject to the constraints
• 4x 2y ? 16,
• 3x 3y ? 15,
• x ? 0,
• y ? 0.
• We graph the system of inequalities and
  determine the vertices.
• Next, we evaluate the objective function P at
  each vertex.

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Example continued
The baker will make a maximum profit when 3 trays
of corn muffins and 2 trays of pumpkin muffins
are produced.
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Example

• Omar owns a car and a moped. He can afford 12
  gal of gasoline to be split between the car and
  the moped. Omars car gets 20 mpg and, with the
  fuel currently in the tank, can hold at most an
  additional 10 gal of gas. His moped gets 100 mpg
  and can hold at most 3 gal of gas.
• How many gallons of gasoline should each vehicle
  use if Omar wants to travel as far as possible?
• What is the maximum number of miles that he can
  travel?

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Example continued

• Omar owns a car and a moped. He can afford 12
  gal of gasoline to be split between the car and
  the moped. Omars car gets 20 mpg and, with the
  fuel currently in the tank, can hold at most an
  additional 10 gal of gas. His moped gets 100 mpg
  and can hold at most 3 gal of gas.

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