Graph linear inequalities on the coordinate plane.

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Objectives
Graph linear inequalities on the coordinate
plane. Solve problems using linear inequalities.
Linear functions form the basis of linear
inequalities. A linear inequality in two
variables relates two variables using an
inequality symbol, such as y gt 2x 4. Its graph
is a region of the coordinate plane bounded by a
line. The line is a boundary line, which divides
the coordinate plane into two regions.
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For example, the line y 2x 4, shown at
right, divides the coordinate plane into two
parts one where y gt 2x 4 and one where y lt 2x
4. In the coordinate plane higher points have
larger y values, so the region where y gt 2x 4
is above the boundary line where y 2x 4.
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To graph y 2x 4, make the boundary line
solid, and shade the region above the line. To
graph y gt 2x 4, make the boundary line dashed
because y-values equal to 2x 4 are not included.
4
Graph the inequality .
Draw the boundary line dashed because it is not
part of the solution.
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Example 1A Continued
Check Choose a point in the solution region, such
as (3, 2) and test it in the inequality.
The test point satisfies the inequality, so the
solution region appears to be correct.
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Graph the inequality y 1.
Recall that y 1 is a horizontal line.
Check The point (0, 2) is a solution because 2
1. Note that any point on or below y 1 is
a solution, regardless of the value of x.
.
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Graph the inequality y 3x 2.
The boundary line is y 3x 2 which has a
yintercept of 2 and a slope of 3.
Draw a solid line because it is part of the
solution. Then shade the region above the
boundary line to show y gt 3x 2.
Check Choose a point in the solution region, such
as (0, 0) and test it in the
inequality.
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If the equation of the boundary line is not in
slope-intercept form, you can choose a test point
that is not on the line to determine which region
to shade. If the point satisfies the inequality,
then shade the region containing that point.
Otherwise, shade the other region.
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Graph 3x 4y 12 using intercepts.
Step 1 Find the intercepts.
Substitute x 0 and y 0 into 3x 4y 12 to
find the intercepts of the boundary line.
y-intercept
x-intercept
3x 4y 12
3x 4y 12
3(0) 4y 12
3x 4(0) 12
4y 12
3x 12
y 3
x 4
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Step 2 Draw the boundary line. The line goes
through (0, 3) and (4, 0). Draw a solid line for
the boundary line because it is part of the graph.
Step 3 Find the correct region to
shade. Substitute (0, 0) into the inequality.
Because 0 0 12 is true, shade the region that
contains (0, 0).
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Graph 3x 4y gt 12 using intercepts.
(4, 0)
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Many applications of inequalities in two
variables use only nonnegative values for the
variables. Graph only the part of the plane that
includes realistic solutions.
A school carnival charges 4.50 for adults and
3.00 for children. The school needs to make at
least 135 to cover expenses.
A. Using x as the adult ticket price and y as
the child ticket price, write and graph an
inequality for the amount the school makes
on ticket sales.
B. If 25 child tickets are sold, how many adult
tickets must be sold to cover expenses?
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Let x represent the number of adult tickets and y
represent the number of child tickets that must
be sold. Write an inequality to represent the
situation.
An inequality that models the problem is 4.5x
3y 135.
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Find the intercepts of the boundary line.
4.5x 3(0) 135
4.5(0) 3y 135
y 45
x 30
Graph the boundary line through (0, 45) and (30,
0) as a solid line. Shade the region above the
line that is in the first quadrant, as ticket
sales cannot be negative.
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If 25 child tickets are sold,
4.5x 3(25) 135
Substitute 25 for y in 4.5x 3y 135.
4.5x 75 135
Multiply 3 by 25.
A whole number of tickets must be sold.
At least 14 adult tickets must be sold.
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ADVANCED LEVEL
Solve for y. Graph the
solution.
Solve 2(3x 4y) gt 24 for y. Graph the solution.