Q. 406 - Intermediate Algebra

Question

Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.

Annie has a budget of $\backslash (600$ to purchase
paperback books and hardcover books for her
classroom. She wants the number of hardcover to be at least $5$ more than three times the number of paperback books. Paperback books cost $\backslash )4$ each and hardcover
books cost $\$15$ each.
(a) Write a system of inequalities to model this
situation.
(b) Graph the system.
(c) Can she buy $8$ paperback books and $40$ hardcover books?
(d) Can she buy $10$ paperback books and $37$ hardcover books?

Step-by-Step Solution

Answer

(a) The system of inequality to model the situation is $\left\{\begin{array}{l}y\ge 3x+5\\ 4x+15y\le 600\end{array}\right.$

(b) The graph of the inequality is the overlapped region in the graph:

(c) No, she cannot buy $8$ paperback books and $40$ hardcover books.

(d) Yes, she can buy $10$ paperback books and $37$ hardcover books

Part (a) Step 1. Given

Annie has a budget of $\$600$

She wants the number of hardcovers to be at least $5$ more than three times the number of paperback books.

Paperback books cost $\$4$ each and hardcover books cost $\$15$ each.

Part (a) Step 2. Form linear inequality

Let $x$ denote the number of paperback books and

$y$ denote the number of hardcover books.

She wants the number of hardcover to be at least $5$ more than three times the number of paperback books.

So, $y\ge 3x+5$

Paperback books cost $\$4$ each and hardcover books cost $\$15$ each.

$4x+15y\le 600$

Part (b) Step 1. Graph the first inequality

Graph the inequality $y=3x+5$

The boundary line is a solid line since the inequality is $\ge$.

Test $(0,0)$ and it is false so shade the region that does not contain the test point.

Part (b) Step 2. Graph the second inequality

Graph the equation $4x+15y=600$

The boundary line is a solid line since the inequality sign is $\le$.

Test $(0,0)$ and it is true so shade the region that contains the test point.

Part (b) Step 3. Check the inequality

The solution of the inequality is the double shaded region.

Choose any point from the double shaded region and check the inequality

Choose $(0,20)$

Check the first inequality

$y\ge 3x+5$

$20\ge 3\left(0\right)+5$

$20\ge 5$ is true.

Check the second inequality: 

      $4x+15y\le 600$

$4\left(0\right)+15\left(20\right)\le 600$

               $300\le 600$ is true.

Part (c) Step 1. Given

Can she buy $8$ paperback books and $40$ hardcover books?

Part (c) Step 2. Substitute the points in inequalities

Substitute $x=8$ and

                  $y=40$ in the inequalities,

$40\ge 3\left(8\right)+5$

$40\ge 29$ is true.

Then $4x+15y\le 600$

  $4\left(8\right)+15\left(40\right)\le 600$

                 $632\le 600$ is false.

Since one of the inequality is false, she cannot buy $8$ paper books and $40$hardcover books.

Part (d) Step 1. Given

Can she buy $10$ paperback books and $37$ hardcover books?

Part (d) Step 2. Substitute the points in inequalities

Substitute $x=10$,

                  $y=37$ in the inequality,

$y\ge 3x+5$

$37\ge 3\left(10\right)+5$

$37\ge 35$ is true.

Then, $4x+15y\le 600$

  $4\left(10\right)+15\left(37\right)\le 600$

                   $595\le 600$ is true.

Since both the inequalities are true, she can buy $10$ paper books and $37$ hardcover books.