6.5 Graphing Linear Inequalities in Two Variables 7.6 Graphing Linear Inequalities System

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6.5 Graphing Linear Inequalities in Two
Variables7.6 Graphing Linear Inequalities System

• Objectives
• 1. Graph linear inequalities.
• 2. Graph systems of linear inequalities.

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Linear Inequalities

• A linear inequality in two variables is an
  inequality that can be written in the form
• Ax By lt C,
• where A, B, and C are real numbers and A and B
  are not both zero. The symbol lt may be replaced
  with ?, gt, or ?.
• 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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To Graph a Linear Inequality

• Step
• (1) Solve for y, convert the inequalities to
  Slope-Intercept Form. If one variable is
  missing, solve it and go to step (2).
• (2) Graph the related equation.
• If the inequality symbol is lt or gt, draw the
  line dashed.
• If the inequality symbol is ? or ?, draw
  the line solid.
• (3) If y lt or y ? , shade the region BELOW the
  line
• If y gt or y ? , shade the region ABOVE
  the line
• If x lt or x ? , shade the region LEFT to
  the line
• If x gt or x ? , shade the region RIGHT to
  the line

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Example

• Graph y gt x ? 4.
• Already in S-I form.
• The related equation is y x ? 4. Use a
  dashed line because the inequality symbol is gt.
  This indicates that the line itself is not in the
  solution set.
• (3) Determine which half-plane satisfies the
  inequality.
• y gt x ? 4
• gt shade above

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Example

• Graph 4x 2y ? 8
• 1. Convert to S-I form
• y ? -2x 4
• 2. Graph the related equation.
• 3. Determine which half-plane satisfies the
  inequality.
• y ? shade below

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Example

• Graph 2x 4 gt 0 .
• 1. One variable is missing. Solve it
• x gt 2
• 2. Graph the related equation
• x 2
• 3. Determine which half-plane satisfies the
  inequality.
• x gt shade the right.

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Example

• Graph 8 - 3y ? 2 .
• 1. One variable is missing. Solve it
• y ? 2
• 2. Graph the related equation
• y 2
• 3. Determine the region to be shaded.
• y ? shaded below

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7.6 Systems of Linear Inequalities

• Graph the solution set of the system.
• First, we graph x y ? 3 using a solid line.
• y ? - x 3
• above
• Next, we graph x ? y gt 1 using a dashed line.
• y lt x 1
• below

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
• We graph the related equations using solid
  lines. We shade the region common to all three
  solution sets.

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Example continued

• 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).

• 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).

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• Summary
• To graph a two-variable linear inequality, graph
  the related equation first (variable y must be
  solved) with appropriate boundary line
• lt and gt use dash line
• and use solid line
• y lt and y shade the region below
• y gt and y shade the region above
• x lt and x shade the region left
• x gt and x shade the region right
• When graphing the linear inequality system,
  follow the step 1 3 and choose the region
  shaded most.

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• Assignment
• 6.5 P 363 s 10 - 22 (even), 32 - 40 (even), 45
  - 60 (even)
• 7.6 P 435 s 9 - 26