Solving systems of inequalities

Question

A store has two types of animal feed available. Type A contains 5

 pounds of oats and 7

 pounds of corn per bag. Type B contains 6

 pounds of oats and 4

 pounds of corn per bag. A farmer buys both types and combines them so that the resulting mixture has at least 150

 pounds of oats and at least 140

 pounds of corn. The store only has 20

 bags of type A feed and 25

 bags of type B feed in stock. Let x

 be the number of type A bags purchased. Let y

 be the number of type B bags purchased. Shade the region corresponding to all values of x

 and y

 that satisfy these requirements

Answer 1

So x is the number of Type A bags purchased, and y is the number of Type B bags purchased.

Type A contains 5 pounds of oats and 7 pounds of corn per bag.

Type B contains 6 pounds of oats and 4 pounds of corn per bag.

The farmer wants a mixture containing at least 150 pounds of oats and at least 140 pounds of corn.

The word "at least" implies "greater than or equal to."

Therefore, the inequality for pounds of oats can be written as

5x + 6y ≥ 150

The inequality for pounds of corn can be written as

7x + 4y ≥ 140

To graph these inequalities, treat them like you would linear equations. Rewrite each "equation" in slope-intercept form. Remember that slope-intercept form is y = mx + b.

5x + 6y ≥ 150 --> 5x + 6y = 150 --> y = -5/6x + 25

7x + 4y ≥ 140 --> 7x + 4y = 140 --> y = -7/4x + 35

Plot each "equation" using the slope and the x- and y-intercepts.

Be sure to use solid boundary lines for both inequalities, as you are dealing with an inclusive symbol for both (i.e., "greater than or equal to"). When you graph the inequalities, the area you shade will be the one containing solutions that apply to both inequalities.