a A rectangle has two dimensions, use them as your variables. To enclose a rectangle, you need to know its perimeter. What can you get from this?
B
b What does it mean when an equation involves only one variable? Remember, when an inequality is notstrict, the line is solid. Can there be negative values in the solution?
A
a x≤ 30 2x+2y ≤ 180
B
bGraph:
Practice makes perfect
a Since the garden has a rectangular form, it has two dimensions: width and length. Hence, we can define two variables, one for each dimension.
&x =width of the rectangular garden
&y =length of the rectangular garden
With these variables defined, we can imagine that the garden looks as shown below.
Since we are told that "the garden can be no more than 30 ft wide," x must be less than or equal to 30. From this, we get our first inequality.
x ≤ 30
On the other hand, in order to enclose the entire garden, the farmer will need 2x+2y ft of chicken wire. But, remember that he "would like to use at most 180 ft of chicken wire," which implies that the perimeter of the rectangle must be less than or equal to 180 ft. Therefore, we have our second inequality.
2x+2y ≤ 180
In conclusion, the system of linear inequalities that models the given situation is as shown below.
x≤ 30 2x+2y ≤ 180
b Let's graph the system of inequalities that we created in Part A. We can see that the first inequality involves only the x-variable while the second one involves both x and y-variables.
Graphing Inequality (I)
The equation of the boundary line is found by replacing the inequality symbol with an equals sign.
Inequality & Boundary Line
x ≤ 30 & x =30
We can see that this is a vertical line whose x-intercept is 30. Additionally, since the inequality is notstrict, the boundary line will be solid. The region represented by x ≤ 30 is all the points that are to the left of the boundary line.
Graphing Inequality (II)
We will find the boundary line in the same way for this inequality as well.
Inequality & Boundary Line
2x+2y ≤ 180 & 2x+2y = 180
The boundary line is not in slope-intercept form. Let's rewrite it.
From this last equation, we can see that the slope of the boundary line is - 1 and the y-intercept is 90. Also, since the initial inequality is not strict (≤), the line will be solid. Let's plot it in the previous coordinate plane.
Now, let's test the origin in the second inequality to determine which region we have to shade. If the point (0,0) produces a true statement, we will shade the region containing it. Otherwise, we will shade the other side of the line.
Since 0 is less than or equal to 180, we've got a true statement. Hence, we will shade the region that contains the origin.
Final Graph
Finally, the graph that represents all possible solutions is the overlapping part. But, remember that x and y both represent the dimensions of the garden, so they have to be positive. Therefore, we will limit the region only to the first quadrant.
