Karen is buying pens and pencils for the new school year. she wants to have no more than 25 writing utensils in all. she also wants the number of pencils to be greater than or equal to the square of 3 less than the number of pens. create a system of inequalities to model the situation above, and use it to determine how many of the solutions are viable.

Let x be the number of pens she buys and y be the number of pencils she buys.  Our first inequality is
x + y ≤ 25.  When we graph this, it might be helpful to have y isolated.  To do this, subtract x from both sides:
x + y - x ≤ 25 - x
y ≤ 25 - x
Our second inequality is
y ≥ (x - 3)²
This is the same as y ≥ (x - 3)(x - 3); multiplying we have y ≥ x² - 3x - 3x + 9 or y ≥ x² - 6x + 9.
Graphing these in the graphing calculator will give you the graph pictured in the screenshots.  We can see that the minimum value for x (pens) would have to be 0, in which case she could purchase 25 pencils.  The maximum value for x that works would be 7, in which case she could purchase 18 pencils.