is capable of carrying 35 pounds while Marc can
carry 50 pounds.
a. Graph the inequalities that represent how much
they can carry.
b. How many days can they camp, assuming that
they bring all their supplies in at once?
c. Who will run out of supplies first?
3-2 Solving Systems of Inequalities by Graphing
Solve each system of inequalities by graphing.
SOLUTION: a. A gallon of water weighs approximately 8 pounds,
so 0.5 gallon of water weighs about 4 pounds. Let x
be the number of days and y be the number of
pounds.
The system of inequalities that represent the situation
is:
15. SOLUTION: Graph the system of inequalities in a coordinate
plane.
Graph the system of inequalities in the same
coordinate plane.
Find the coordinates of the vertices of the
triangle formed by each system of inequalities.
19. SOLUTION: Graph the system of inequalities.
b & c. Assume Jessica only uses the supplies she
has carried, and Marc only uses the supplies he has
carried. Then Marc will run out of supplies at the
point where y = 9x + 20 intersects the line y = 50. The coordinates of the vertices are (2, –1),
(5, 8), and (–7, 8).
27. CCSS REASONING On a camping trip, Jessica
needs at least 3 pounds of food and 0.5 gallon of
water per day. Marc needs at least 5 pounds of food
and 0.5 gallon of water per day. Jessica’s equipment
weighs 10 pounds, and Marc’s equipment weighs 20
pounds.
A gallon of water weighs approximately 8 pounds.
Each of them carries their own supplies, and Jessica
is capable of carrying 35 pounds while Marc can
carry 50 pounds.
a. Graph the inequalities that represent how much
they can carry.
b. How many days can they camp, assuming that
they bring all their supplies in at once?
c. Who will run out of supplies first?
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SOLUTION: a. A gallon of water weighs approximately 8 pounds,
This point is
supplies after
, so Marc will run out of
days. Jessica will run out of supplies at the point where y = 7x + 10 intersects the
line y = 35. This point is
, so Jessica will
run out of supplies after 3.57 days. Therefore, Marc will run out of supplies first; Jessica can last about a
quarter of a day longer.
Solve each system of inequalities by graphing.
29. SOLUTION: Graph the system of inequalities.
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supplies at the point where y = 7x + 10 intersects the
line y = 35. This point is
, so Jessica will
run out of supplies after 3.57 days. Therefore, Marc will run Systems
out of supplies
first; Jessica
last about a
3-2 Solving
of Inequalities
bycan
Graphing
quarter of a day longer.
SOLUTION: Graph the inequalities in the same coordinate plane.
Solve each system of inequalities by graphing.
29. SOLUTION: Graph the system of inequalities.
35. SOLUTION: Graph the system of inequalities.
The area defined by the inequalities is the region
quadrilateral ABCD.
The coordinates A(2, 6), B(5, 0), C(-3, -4), and D(-6,
8) can be determined from the graph. To find the
area, divide the region into
,
and trapezoid AECF by drawing lines
.
and 45. CHALLENGE Find the area of the region defined
by the following inequalities.
To find the area of each shape use the appropriate
area formula.
SOLUTION: Graph the inequalities in the same coordinate plane.
Find the height by using the length of
where K is
(2, 0). This height is 3 units long. The corresponding base is
or 7.5.
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3-2 Solving Systems of Inequalities by Graphing
Find the height by using the length of
where G is
(-3, 8). This height is 3 units long. The corresponding
base is
or 11.25.
Trapezoid AECF
Let
,
and .
First determine the related equations. To do this,
analyze the graph to find the slope and one point for
each line. The uppermost boundary has a slope of
or –0.5.
This line has a y-intercept of 4 so the equation is y =
–0.5x + 4. Since the line is solid and the shading is
below the line, the inequality is
.
The boundary to the left of the origin has a slope of
or -3. The y-intercept is -6 so the equation is y =
–3x – 6. Since the line is solid and the shading is
above the line, the inequality is
.
The boundary to the right of the origin has a slope of
or 2. This line has a y-intercept of -6 so the
equation is y = 2x – 6. Since the line is solid and the
shading is above the line, the inequality is
.
To determine how many points with integer
coordinates are solutions to the system, analyze the
graph. Redraw the graph with each axis scaled by
1. Count every integer ordered pair on the
boundary lines as well as all of the integer points
within the area. Add the areas of the three figures together to find
the area of the enclosed figure.
Therefore, the area of the enclosed figure is 75
square units.
47. CHALLENGE Write a system of inequalities to
represent the solution shown. How many points with
integer coordinates are solutions of the system?
First count all of the ordered pairs that lie on the
boundary lines. There are 12 points. Next count the
ordered pairs on the axes to find that there are 12.
There are 7 points in the first quadrant, 10 in
Quadrant II, 2 in Quadrant III, and 4 in Quadrant IV.
There are a total of 47 points with integer
coordinates that are solutions to the system.
SOLUTION: Sample answer:
First determine the related equations. To do this,
analyze the graph to find the slope and one point for
each line. The uppermost boundary has a slope of
or –0.5.
This line has a y-intercept of 4 so the equation is y =
+ 4. Since
thebyline
is solid and the shading is
–0.5x
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below the line, the inequality is
.
The boundary to the left of the origin has a slope of
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