Warm Up

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Warm Up
Problem of the Day
Lesson Presentation
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Warm Up Find each equation of direct variation,
given that y varies directly with x. 1. y is 18
when x is 3. 2. x is 60 when y is 12. 3. y is 126
when x is 18. 4. x is 4 when y is 20.
y 6x
y 7x
y 5x
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Problem of the Day The circumference of a pizza
varies directly with its diameter. If you graph
that direct variation, what will the slope be?
?
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Learn to graph inequalities on the coordinate
plane.
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Insert Lesson Title Here
Vocabulary
boundary line linear inequality solution
set closed half-plane open half-plane
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A graph of a linear equation separates the
coordinate plane into three parts the points on
one side of the line, the points on the boundary
line, and the points on the other side of the
line.
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When the equality symbol is replaced in a linear
equation by an inequality symbol, the statement
is a linear inequality. Any ordered pair that
makes the linear inequality true is a solution.
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Additional Example 1A Graphing Inequalities
Graph each inequality. y lt x 1
First graph the boundary line y x 1. Since no
points that are on the line are solutions of y lt
x 1, make the line dashed. Then determine on
which side of the line the solutions lie.
(0, 0)
Test a point not on the line.
y lt x 1
Substitute 0 for x and 0 for y.
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Additional Example 1A Continued
Since 0 lt 1 is not true, (0, 0) is not a
solution of y lt x 1. Shade the side of the line
that does not include (0, 0).
(0, 0)
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Additional Example 1B Graphing Inequalities
y ? 2x 1
First graph the boundary line y 2x 1. Since
points that are on the line are solutions of y ?
2x 1, make the line solid. Then shade the part
of the coordinate plane in which the rest of the
solutions of y ? 2x 1 lie.
(0, 4)
Choose any point not on the line.
Substitute 0 for x and 4 for y.
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Additional Example 1B Continued
Since 4 ? 1 is true, (0, 4) is a solution of y ?
2x 1. Shade the side of the line that includes
(0, 4).
(0, 4)
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Additional Example 1C Graphing Inequalities
2y 5x lt 6
First write the equation in slope-intercept form.
2y 5x lt 6
2y lt 5x 6
Subtract 5x from both sides.
Divide both sides by 2.
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Additional Example 1C Continued
(0, 0)
Choose any point not on the line.
Substitute 0 for x and 0 for y.
(0, 0)
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Check It Out Example 1A
Graph each inequality. y lt x 4
First graph the boundary line y x 4. Since no
points that are on the line are solutions of y lt
x 4, make the line dashed. Then determine on
which side of the line the solutions lie.
(0, 0)
Test a point not on the line.
y lt x 4
Substitute 0 for x and 0 for y.
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Check It Out Example 1A Continued
Since 0 lt 4 is not true, (0, 0) is not a
solution of y lt x 4. Shade the side of the line
that does not include (0, 0).
(0, 0)
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Check It Out Example 1B
y gt 4x 4
First graph the boundary line y 4x 4. Since
points that are on the line are solutions of y ?
4x 4, make the line solid. Then shade the part
of the coordinate plane in which the rest of the
solutions of y ? 4x 4 lie.
(2, 3)
Choose any point not on the line.
Substitute 2 for x and 3 for y.
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Check It Out Example 1B Continued
Since 3 ? 12 is not true, (2, 3) is not a
solution of y ? 4x 4. Shade the side of the
line that does not include (2, 3).
(2, 3)
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Check It Out Example 1C
3y 4x ? 9
First write the equation in slope-intercept form.
3y 4x ? 9
3y ? 4x 9
Subtract 4x from both sides.
Divide both sides by 3.
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Check It Out Example 1C Continued
(0, 0)
Choose any point not on the line.
Substitute 0 for x and 0 for y.
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Additional Example 2 Career Application
A successful screenwriter can write no more than
seven and a half pages of dialogue each day.
Graph the relationship between the number of
pages the writer can write and the number of
days. At this rate, would the writer be able to
write a 200-page screenplay in 30 days?
First find the equation of the line that
corresponds to the inequality.
In 0 days the writer writes 0 pages.
point (0, 0)
point (1, 7.5)
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Additional Example 2 Continued
With two known points, find the slope.
y ? 7.5 x 0
The y-intercept is 0.
Graph the boundary line y 7.5x. Since points on
the line are solutions of y ? 7.5x make the line
solid. Shade the part of the coordinate plane in
which the rest of the solutions of y ? 7.5x lie.

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Additional Example 2 Continued
(2, 2) Choose any point not on the
line.
y ? 7.5x
Substitute 2 for x and 2 for y.
Since 2 ? 15 is true, (2, 2) is a solution of y ?
7.5x. Shade the side of the line that includes
point (2, 2).
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Additional Example 2 Continued
The point (30, 200) is included in the shaded
area, so the writer should be able to complete
the 200 page screenplay in 30 days.
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Check It Out Example 2
A certain author can write no more than 20 pages
every 5 days. Graph the relationship between the
number of pages the writer can write and the
number of days. At this rate, would the writer be
able to write 140 pages in 20 days?
First find the equation of the line that
corresponds to the inequality.
In 0 days the writer writes 0 pages.
point (0, 0)
In 5 days the writer writes no more than 20 pages.
point (5, 20)
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Check It Out Example 2 Continued
With two known points, find the slope.
The y-intercept is 0.
y ? 4x 0
Graph the boundary line y 4x. Since points on
the line are solutions of y ? 4x make the line
solid. Shade the part of the coordinate plane in
which the rest of the solutions of y ? 4x lie.
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Check It Out Example 2 Continued
(5, 60) Choose any point not on
the line.
y ? 4x
Substitute 5 for x and 60 for y.
Since 60 ? 20 is not true, (5, 60) is not a
solution of y ? 4x. Shade the side of the line
that does not include (5, 60).
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Check It Out Example 2 Continued
y
200 180 160 140 120 100
80 60 40\ 20
Pages
x
5 10 15 20 25 30 35 40 45 50
Days
The point (20, 140) is not included in the shaded
area, so the writer will not be able to write 140
pages in 20 days.
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Lesson Quiz Part I
Graph each inequality. 1. y lt x 4
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Lesson Quiz Part II
2. 4y 2x gt 12
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Lesson Quiz Part III
Tell whether the given ordered pair is a solution
of each inequality. 3. y lt x 15 (2, 8) 4. y ?
3x 1 (7, 1)
yes
no