Question 1205786
using the desmos.com calculator, you would graph the opposite of the constraint inequalities.


the area that is not shaded is the region of feasibility.


if the inequality is < or >, then any point on the line is not feasible.


if the inequality is <= or >=, then any point on the line is feasible.


all points in the area that is not shaded are feasible.


in these inequalities, the dependent variable is y and the independent variable is x.


you select the value of x and the value of y is then determined from the equation.


your inequalities are:


x < 22 - x
y <= x/3
x >= 0
y >= 0


x is the number of hours for algebra.
y is the number of hours for chemistry.


the first one says that your total number of hours available is less than 22.
the equation starts off as x + y < 22 and you solve for y to get y < 22 - x.
you would draw that line as dashed, since the value cannot be 22.


the second one says that the hours to study algebra are at least as 3 times the hours to study chemistry.
you start with x >= 3y
you solve for y to get y <= x/3


the third and fourth simply state that the number of hours can't be negative.


you are solving for y because you want y to be the dependent variable.
this means you pick the number of hours and the equation tells you whether those number of hours are valid, given the constraints.


the graph looks like this.


<img src = "http://theo.x10hosting.com/2024/012502.jpg">


the points (14,9) and (15,9) are both not valid because they are outside the region of feasibility.


applying the inequalities to these points gets you:


(14,9) inequality of x + y = 23 which is not less than 22.
(15,9) inequality of x + y = 24 which is also not less than 22.
in addition, y <= x/3 gets you y = 14/3 = 4.67 and 9 is not less than or equal to that.
similarly, y <= x/3 gets you y = 15/3 = 5 and 9 is also not less than or equal to that.


bottom line, the two points fail on both constraints.


(9,3) is valid because the sum is less than 22 and 3 <= 9/3.
(13,3) is valid because the sum is less than 22 and 3 <= 13/3.


(19,3) is not valid because the sum is 22 which is not less than 22.
3 is <= 19/3, so that test is passed.
unfortunately all tests need to be passed for the point to be valid, not just some of them.


when you draw the graph manually, you would shade the region that is not shaded on the desmos.com graph.


in other words, you would graph he inequalities as is and not the reverse of the inequalities.


in desmos, the reverse of the inequalities is graphed because the region of feasibility shows up clearly.
it would not if you tried to graph the inequalities as they are using desmos.


the calculator can be found at <a href = "https://www.desmos.com/calculator" target = "_blank">https://www.desmos.com/calculator</a>


let me know if you have any questions.
theo