Systems of Inequalities

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Systems of Inequalities

• Lesson 6.5

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• Frequently, real-world situations involve a range
  of possible values. Algebraic statements of these
  situations are called inequalities.

3

• Recall that you can perform operations on
  inequalities very much like you do on equations.
• You can add or subtract the same quantity on both
  sides, multiply by the same number or expression
  on both sides, and so on.
• The one exception to remember is that when you
  multiply or divide by a negative quantity or
  expression, the inequality symbol reverses.

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• In this lesson you will learn how to graphically
  show solutions to inequalities with two
  variables, such as the last two statements in the
  table above.

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Paying for College

• A total of 40,000 has been donated to a college
  scholarship fund. The administrators of the fund
  are considering how much to invest in stocks and
  how much to invest in bonds. Stocks usually pay
  more but are often a riskier investment, whereas
  bonds pay less but are usually safer.
• Let x represent the amount in dollars invested in
  stocks, and let y represent the amount in dollars
  invested in bonds. Graph the equation x y
  40,000.

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• Name at least five pairs of x- and y-values that
  satisfy the inequality x y lt 40,000 and plot
  them on your graph. In this problem, why can x
  y be less than 40,000?

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• Describe where all possible solutions to the
  inequality x y lt 40,000 are located. Shade this
  region on your graph.

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• Describe some points that fit the condition x
  y 40,000 but do not make sense for the
  situation.

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• Assume that each optionstocks or bondsrequires
  a minimum investment of 5,000, and that the fund
  administrators want to purchase some stocks and
  some bonds. Based on the advice of their
  financial advisor, they decide that the amount
  invested in bonds should be at least twice the
  amount invested in stocks.
• Translate all of the limitations, or constraints,
  into a system of inequalities. A table might help
  you to organize this information.

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Assume that each optionstocks or bondsrequires
a minimum investment of 5,000, and that the fund
administrators want to purchase some stocks and
some bonds. Based on the advice of their
financial advisor, they decide that the amount
invested in bonds should be at least twice the
amount invested in stocks.
Assume that each optionstocks or bondsrequires
a minimum investment of 5,000, and that the fund
administrators want to purchase some stocks and
some bonds. Based on the advice of their
financial advisor, they decide that the amount
invested in bonds should be at least twice the
amount invested in stocks.
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• Graph all of the inequalities and determine the
  region of your graph that will satisfy all the
  constraints. Find each corner, or vertex, of this
  region.

A (5000, 35000)
B (13,333, 26,666.67)
C (5000, 10000)
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• When there are one or two variables in an
  inequality, you can represent the solution as a
  set of ordered pairs by shading the region of the
  coordinate plane that contains those points.

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• When you have several inequalities that must be
  satisfied simultaneously, you have a system.
• The solution to a system of inequalities with two
  variables will be a set of points. This set of
  points is called a feasible region.
• The feasible region can be shown graphically as
  part of a plane, or sometimes it can be described
  as a geometric shape with its vertices given.

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Example A

• Rachel has 3 hours to work on her homework
  tonight. She wants to spend more time working on
  math than on chemistry, and she must spend at
  least a half hour working on chemistry.
• Let x represent time in hours spent on math, and
  let y represent time in hours spent on chemistry.
  Write inequalities to represent the three
  constraints of the system.

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Example A

• Rachel has 3 hours to work on her homework
  tonight. She wants to spend more time working on
  math than on chemistry, and she must spend at
  least a half hour working on chemistry.
• Graph your inequalities and shade the feasible
  region.

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Example A

• Rachel has 3 hours to work on her homework
  tonight. She wants to spend more time working on
  math than on chemistry, and she must spend at
  least a half hour working on chemistry.
• Find the coordinates of the vertices of the
  feasible region.

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Example A

• Name two points that are solutions to the system,
  and describe what they mean in the context of the
  problem.

The points (1.5, 1) and (2.5, 0.5) are two
solutions to the system. Every point in the
feasible region represents a way that Rachel
could divide her time. The solution point (1.5,
1) means she could spend 1.5 h on mathematics and
1 h on chemistry and still meet all her
constraints. The point (2.5, 0.5) means that
Rachel could spend 2.5 h on math and 0.5 h on
chemistry. This point represents the boundaries
of two constraints She cant spend less than 0.5
h on chemistry or more than 3 h total on homework.
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Example B

• Anna throws a ball straight up next to a
  building. The balls height in feet after t
  seconds is given by -16t251t3. Tom rides a
  glass elevator down the outside of the same
  building. His height from the time Anna throws
  the ball can be expressed as 43-5t. As Tom is
  riding down, he sees a bird fly by above the
  elevator but below the ball. When did Tom see the
  bird? Give a range of possible times.

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• At the time Tom sees the bird, the birds height,
  h, must satisfy 43-5tlthlt -16t251t3.
• You can graph these two inequalities separately.
  You might find it easier to graph the boundaries
  first and then shade the feasible region.

Tom saw the bird between 1 and 2.5 seconds after
Anna threw the ball.