Activity 3 - 5

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Activity 3 - 5

• Will Trees Grow?

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Objectives

• Graph a linear inequality in two variables
• Solve a system of linear inequalities in two
  variables graphically
• Determine the corner points of the solution set
  of a system of linear inequalities

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Vocabulary

• Half-plane the set of all points on one side of
  a line.
• System of linear inequalities two or more
  linear inequalities in two variables.
• Corner points the points determined by the
  intersection of the boundary lines of the graph
  of a system of linear inequalities.

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Graphing Inequalities

• Points to Remember
• When the inequality is or , then the line is
  included and is drawn as a solid line (equal
  included)
• When the inequality is lt or gt, then the line is
  not included and is drawn as a dashed line (line
  not included)
• All the area on one side of the line is feasible
  (meets the inequality constraint)
• All the area on the other side of the line is
  infeasible (does not satisfy the inequality
  constraint)
• If inequalities can be reduced to a
  slope-intercept form, then your calculator can
  help solve the problem

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Inequalities Slope-Intercept Form

• Inequalities written in slope-intercept form
• y lt mx b or y mx b Below the line is a
  shaded region
• y gt mx b or y mx bAbove the line is a
  shaded region
• Solution of a system of inequalities is the
  intersection (if any) of all solution regions
  (the shaded region)The intersection of the
  boundary lines of the shaded region is called a
  corner point

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If All Else Fails .

• Remember, we can plug in a point, above or below
  the line, and see if it satisfies the inequality.
  If it works, then that side of the line is
  shaded and if it doesnt work, then the other
  side of the line is shaded.

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Graphical Method Solutions Space

• Consistent
  Inconsistent
  (Parallel Lines)Solution Space
  No Solution Solution Space

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TI-84 Help

• Change inequalities to an equal sign and solve
  for y (y mx b format)
• Place equations into Y1 and Y2
• The shade regions belowPress Y and move cursor
  to the extreme leftPress ENTER as many times as
  necessary to change the icon to
• The shade regions abovePress Y and move cursor
  to the extreme leftPress ENTER as many times as
  necessary to change the icon to
• Solution Space is the cross-hatched region

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Will Trees Grow?

• While researching a term paper on global climate
  change, you discover a mathematical model that
  gives the relationship between temperature and
  amount of precipitation that is necessary for
  trees to grow. If t represents the average annual
  temperature in F, and p represents the annual
  precipitation in inches, then
• t 35
• 5t - 7p lt 70
• These inequalities form a system of inequalities
  in two variables. The solution of the system is
  the set of all ordered pairs in the form (t, p)
  that make each of the inequalities in the system
  a true statement.

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Will Trees Grow?

• t 35 5t - 7p lt 70
• Will trees grow in a region in which the average
  temperature is 22F and the annual precipitation
  is 30 inches?
• Will trees grow if the average temperature is
  55F and the annual precipitation is 20 inches?
• In Sydney Australia, the average temperature is
  64F and the annual precipitation is 48 inches.
  Will trees grow in Sydney?

No average temperature must be above 35
No 5(55) 7(20) 135 which is greater than 70
Yes 64 is greater than 35 and 5(64)
7(48) -16 which is less than 70
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Graphing the Solution Space

• t 35
• 5t 7p lt 70 ? 5t lt 7p 70 ? t lt (7/5)p 14

Solution Space
Meets TemperatureConstraint
35
Meets Temperature and Precipitation Constraint
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Where do Trees Grow?

• Fill in the following table to see if trees can
  grow there (without mans help)

City Temp Precip (t,p) Conclusion
Baghdad, Iraq 73 6 (73, 6)
Fairbanks, AK 29 14 (29, 14)
Lima, Peru 66 1 (66, 1)
Aden, Yemen 84 1.8 (84, 2)
No No No No
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Steps to Solve Systems of Inequalities

• Given Two Inequalities
• Step 1 Solve for y
• Bring x terms to other side
• Divide through by constant in front of y
• Remember to switch inequality if dividing by a
  negative number
• Step 2 Enter equations into calculator
• Change icons on far left
• Step 3 Graph with an appropriate window
• Solution space is cross-hatched region

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Example Problem 1

• Given 4x 2y 8
  2x 3y lt 6
• Step 1 - 2y 8 4x
  3y lt 6 2x y
  -4 2x y lt 2 (2/3)x
• Step 2 Y1 2X 4 Y2 -
  (2/3)X 2
• Step 3 Graph (seen below)

Note Y1 is a solid line and Y2 is a dashed line
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Example Problem 2

• Given 3x 2y 16
  x 2y 0
• Step 1 2y 16 3x
  -2y 0 x y 8
  (3/2)x y 0 (1/2)x
• Step 2 Y1 -(3/2)X 8 Y2
  (1/2)X 0
• Step 3 Graph (seen below)

Note Y1 is a solid line and Y2 is a solid line
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Summary and Homework

• Summary
• An inequality of the form Ax By lt C, where A
  and B cannot both equal zero, is called a linear
  inequality in two variables. The symbol lt can be
  replaced by gt, or .
• The solution set of a linear inequality in two
  variables x and y is the collection of all
  ordered pairs (x, y) whose coordinates satisfy
  the given inequality.
• The solution set of a system of linear
  inequalities in two variables is the collection
  of all ordered pairs whose coordinates satisfy
  each linear inequality in the system.
• Homework
• pg 346-352 1, 5-7, 11, 13