91Ó°ÊÓ

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.

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, $yâ‰\yen 3x+5$

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

$4x+15y≤600$

Graph the inequality$y=3x+5$

The boundary line is a solid line since the inequality is $â‰\yen$.

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

Graph the equation$4x+15y=600$

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

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

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

$yâ‰\yen 3x+5$

$20â‰\yen 3\left(0\right)+5$

$20â‰\yen 5$is true.

Check the second inequality:

$4x+15y≤600$

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

$300≤600$ is true.

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

Substitute $x=8$and

$y=40$in the inequalities,

$40â‰\yen 3\left(8\right)+5$

$40â‰\yen 29$is true.

Then $4x+15y≤600$

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

$632≤600$is false.

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

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

Substitute $x=10$,

$y=37$in the inequality,

$yâ‰\yen 3x+5$

$37â‰\yen 3\left(10\right)+5$

$37â‰\yen 35$is true.

Then, $4x+15y≤600$

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

$595≤600$is true.

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