Solving Systems of Inequalities 05:47 minutes

Video Transcript

Transcript Solving Systems of Inequalities

Caleb is very proud of his collection of sneakers and baseball caps. He wants to build a shelving unit to display his treasures. In his garage, he finds 150 nails and 66 wooden planks. Can we help Caleb figure out if there will be enough materials to build the shelving unit?

Solving a system of inequalities

It's no problem at all! To calculate the solution, you can solve a system of inequalities. Caleb knows that, to build a cubby for each baseball cap, he needs 5 nails and 2 wooden planks. Let 'b' equal the number of baseball caps. To build a cubby for each pair of sneakers, he needs 5 nails and 3 wood planks. Let 's' equal the number of pairs of sneakers. Although he has 150 nails and 66 planks to build the shelf, he doesn't have to use them all. Let’s write the two inequalities.
Caleb uses 5 nails for each baseball cap cubby and 5 nails for each sneaker cubby, and the total is less than or equal to 150 nails. He also uses 2 wooden planks for each baseball cap and 3 planks for each pair of sneakers. Caleb can use 66 wooden planks or fewer to construct his cubbies.

Does Caleb have enough materials to build a shelf to display 10 baseball caps and 15 pairs of sneakers? To figure this problem out, let’s substitute 'b' with the number 10 and 's' with the number 15. Simplify by multiplying, and then add. Since both inequalities are true, we know Caleb has enough materials to build the shelving unit. If only one of the inequalities were true, then he would only have enough of one of the materials he needs to build the shelf.
Caleb wants to know if he has enough materials to build cubbies for 5 new pairs of shoes. Instead of 15, we'll substitute the number 20 in our system of equations. He'll have enough nails to complete this project, but he won't have enough wooden planks to finish building the extra cubbies.

Graphing a system of inequalities

We can also figure out the answer to this system of inequalities problem by graphing. To make it easier to graph, let's modify the original equation to let 'x' represent the number of baseball caps and 'y' to represent the number of pairs of sneakers. Next, we need to rewrite the two inequalities in slope-intercept form. To do this, you must isolate the 'y' on the left side of the inequality sign, so subtract the x-term from both sides, and then divide both sides by the coefficient 5 to isolate 'y'. Y' is alone on the left side. Let's do the same to the other equation and the two inequalities are now written in slope-intercept form.

Solving the problem graphically

Now it's much easier to graph each inequality. For the first inequality with a y-intercept equal to 30 and a slope equal to -1, the line and the yellow shaded area of the graph are included in the solution set of the inequality; and for the second inequality with a y-intercept of 22 and a slope equal to -2/3, the line and the red shaded area of the graph are included in the solution set. The orange shaded area of the graph is the overlap of the yellow and red shaded areas, and it’s the solution to this system of inequalities.

Take a look at point 'p', the ordered pair (10, 15), since it’s in the area shaded orange, it’s a possible solution to the system, so we know Caleb has enough nails and planks to build cubbies for 10 baseball caps and 15 pairs of sneakers. On the other hand, point 's', the ordered pair (10, 20), is not in the area shaded orange. From the graph, we can see that Caleb has enough nails, shown by the area shaded yellow, but he does not have enough planks, shown by the area shaded red, to build 10 cubbies for caps and 20 cubbies for sneakers, so point 's' is not a solution.

Although Caleb has enough supplies, he’s not a master with a hammer and, as you can see, the shelving unit looks a little bit unstable he puts the last baseball cap in the cubby, and ...oh no…