For a system of inequalities, each condition shades the pairs that satisfy it. Graph both conditions in Desmos and find the highest point of their overlap to bound the requested variable. Desmos also shades fractional points, so rounding that height down isn't enough: both coordinates of a proposed pair must be integers.
Hints
- Hint 1
The feasible region is where both shadings overlap. To find the greatest possible , look for the highest point in that overlap, not the highest point on either boundary by itself.
- Hint 2
A height below the top of the overlap may still fail. At that height, look across the overlap and ask whether any integer fits between its left and right edges.
Step-by-step
Graph the overlap and check integer points
Step 1Find the highest possible height
Type both inequalities on separate Desmos lines. Each shades the points that satisfy it, so look at their feasible region, where both shadings overlap. Click the point where the solid boundaries meet: Desmos shows . That's the highest point of the overlap, so an integer can be at most .
- Step 2
Check whether the highest integer height works
Add and click its two crossings with the boundaries. Desmos shows and , so a point in the overlap at this height needs . Both coordinates must be integers, not only the one you're maximizing. There's no integer in that interval, so fails.
- Step 3
Find a whole-number point that works
Try the next integer height: add and click where it meets . Desmos shows inside the overlap. That boundary is solid, so the point counts; both coordinates are positive integers. Since fails and no higher integer can work, the greatest possible value of is . Grid in 8.