SOLUTION: 1) Solve the following system of linear inequalities by graphing. 3x � y < 2 x + y > 2 2) A small company produces both bouquets and wreaths of dried flowers. The bouquet

Question 119742: 1) Solve the following system of linear inequalities by graphing.
3x � y < 2
x + y > 2

2) A small company produces both bouquets and wreaths of dried flowers. The bouquets take 1 hour of labor to produce, and the wreaths take 2 hours. The labor available is limited to 80 hours per week, and the total production capacity is 60 items per week. Write a system of inequalities representing this situation, where x is the number of bouquets and y is the number of wreaths. Then graph the system of inequalities.

3) A small company produces both standard and deluxe playhouses. The standard playhouses take 12 hours of labor to produce, and the deluxe playhouses take 20 hours. The labor available is limited to 800 hours per week, and the total production capacity is 50 items per week. Existing orders require the company to produce at least 10 standard playhouses and 15 deluxe playhouses per week. Write a system of inequalities representing this situation, where x is the number of standard playhouses and y is the number of deluxe playhouses. Then graph the system of inequalities.

4) A home-based company produces both hand-knitted scarves and sweaters. The scarves take 2 hours of labor to produce, and the sweaters take 14 hours. The labor available is limited to 40 hours per week, and the total production capacity is 5 items per week. Write a system of inequalities representing this situation, where x is the number of scarves and y is the number of sweaters. Then graph the system of inequalities.

Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
1) Solve the following system of linear inequalities by graphing.
Arrange both equation into the slope/intercept form for graphing
3x � y < 2
-y > -3x + 2
Y has to positive, mult by -1; this reverses the inequality sign
y < 3x - 2
and
x + y > 2
y > -x + 2
:
plot the graphs for x=-4 and x=+4; should look like this:

The shaded area will be above the purple line (y>3x-2) and
Below the green line (y=-x+2)
:
:
2.A small company produces both bouquets and wreaths of dried flowers. The bouquets take 1 hour of labor to produce, and the wreaths take 2 hours. The labor available is limited to 80 hours per week, and the total production capacity is 60 items per week. Write a system of inequalities representing this situation, where x is the number of bouquets and y is the number of wreaths.
:
Let x = no. of bouquets; y = no. of wreaths
:
Labor constraint:
1x + 2y =< 80
2y =< -x + 80
y =< -.5x + 40; divided equation by 2 ( this is the form we want for graphing)
:
Production capacity constraint:
x + y =< 60
y =< -x + 60
:
Then graph the system of inequalities.
Plot two points for each:
y = -.5x + 40
x | y
-------
0 | 40
50 | 15
and
y = -x + 60
x | y
--------
0 | 60
50 | 10
:
Plot and draw these two graphs, should look like this:

Area of feasibility would be at or below the lowest line (Positive values only)
:
:
3.A small company produces both standard and deluxe playhouses. The standard playhouses take 12 hours of labor to produce, and the deluxe playhouses take 20 hours. The labor available is limited to 800 hours per week, and the total production capacity is 50 items per week. Existing orders require the company to produce at least 10 standard playhouses and 15 deluxe playhouses per week. Write a system of inequalities representing this situation, where x is the number of standard playhouses and y is the number of deluxe playhouses. Then graph the system of inequalities.
:
Let x = standard p.h; y = deluxe p.h
:
The labor equation:
12x + 20y =< 800
:
put equation in the general (y=) form to plot the graph:
20y =< 800 - 12x
y =< 800/20 - (12/20)x
y =< 40 - .6x
:
:
The production capacity equation:
x + y =< 50
Put this in the "y=" form also
y <= 50 -x
:
Existing order constraints
x => 10
and
y => 15
:
Plot these using the equation givens.
y = 40 - .6x; (purple line)
y = 50 - x; (green line)
y = 15; Note that y = 15 is a horizontal line going thru y = 15; black line
x = 10 is a vertical line going thru x = 10
:
I assume you know how to make up an x/y table and plot a graph, if you can't,
let me known and we will go thru the graphing routine
:
Here is the graph:

:
1.At or below, the purple line or the green line whichever is lower.
2.At or above the black horizontal line
3.At or to the right of the vertical line
:
:
A home-based company produces both hand-knitted scarves and sweaters. The scarves take 2 hours of labor to produce, and the sweaters take 14 hours. The labor available is limited to 40 hours per week, and the total production capacity is 5 items per week. Write a system of inequalities representing this situation, where x is the number of scarves and y is the number of sweaters. Then graph the system of inequalities.
:
x = no. of scarves; y = no. of sweaters:
:
The labor (in hours) inequality:
2x + 14y =< 40
Arrange in the general (y=) form so we can graph it:
14y =< 40 - 2x
y =< (40/14) - (2/14)x
y =< 20/7 - (1/7)x
:
The production inequality:
x + y = 5
y = 5 - x
:
Plot these two inequalities (I assume you know how to plot a graph, if not you
can email me at ankor@dixie-net.com and I will add the procedure to do that.)
:
your graph should look like this:

:
The area of feasibility would be at or below either line, whichever is lower
It's assumed that x and y => 0

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