juice costs 3 per glass.
cheese costs 6 per pound.
let x = the number of pounds of cheese.
let y = the number of glasses of juice.
if that is all that is being bought is 3000, then the inequality becomes:
6x + 3y <= 3000
subtract 6x from both sides of this inequality to get 3y <= 3000 - 6x.
divide both sides of this inequality by 6 to get y <= 1000 - 2x.
arrange the terms in descending order of degree to get y <= -2x + 1000.
6x + 3y <= 3000 is your inequality in standard form.
y <= -2x + 1000 is your inequality in slope intercept form.
the slope of the inequality is -2.
the y-intercept of the inequality is 1000.
that can be seen easily by looking at the slope intercept form of the inequality.
the slope intercept form of your inequality is y <= mx + b
m is the slope
b is the y-intercept.
since the ineqquality is y <= -2x + 1000, .....
the slope is -2.
the y-intercept is 1000.
the graph of your inequality looks like this.
the number of pounds of cheese are measured on the x-axis.
the number of glasses of juice are measured on the y-axis.
when x = 100, the range of y is between 0 and 800.
when y = 500, the range of x is between 0 and 250.
for example, as shown on the graph.
when y = 500, the maximum value of x is 250.
that buys 250 pounds of cheese and 500 glasses of juice.
total cost is 6 * 250 + 3 * 500 = 3000 dollars.
when x = 100, the maximum value of y is 800.
that buys 100 pounds of cheese and 800 glasses of juice.
total cost is 6 * 100 + 3 * 800 = 3000 dollars.
when y = 500, any value of x between 0 and 250 is valid.
when x = 100, any value of y between 0 and 800 is valid.
since this is an inequality, any value on the graph that is not shaded will give you a valid value for (x,y) coordinate pairs.
for example, when x = 100 and y = 500, you would be buying 100 pounds of cheese and 500 glasses of orange juice for a total cost of 6 * 100 + 3 * 500 = 2100.
since you paid less than 3000, the inequality holds true.
you were asked for the slope of this budget constraint.
the slope is -2.
note that i graphed the opposite of the inequality.
this allows me to see the region of feasibility easier because that becomes the un-shaded part of the graph rather than the shaded part of the graph.
not all graphing calculators allows you to show inequalities or to show linear equations in the form of ax + by = c.
the desmos.com calculator allows this which makes graphing inequalities much easier and automatically generated.