If one number must satisfy two inequalities, you're finding their intersection, the range they share. Enter both inequalities as written in Desmos and read their overlapping shading. Then look for the rightmost whole number, since the question asks for the greatest integer. Don't automatically step left from an endpoint: a solid boundary counts, while a dotted one does not.
Hints
- Hint 1
An inequality allows a range of numbers, not just one value. Type the first inequality into Desmos and read its shaded side. Does its dotted boundary belong to that range?
- Hint 2
Together, the inequalities form a system, so a number must work in both shaded ranges. Add the second inequality on the same graph. Where do the shadings overlap?
- Hint 3
A solid boundary counts because allows equality. Look at the right edge of the shared shading: is that edge itself an integer you can use?
Step-by-step
Graph the shared range in Desmos
Step 1Find the first allowed range
Type into Desmos as written. It shades to the right of the dotted vertical boundary . The strict inequality leaves out equality, so the first condition requires .
- Step 2
Use the right edge of the overlap
Keep the first line and type . Desmos shades to the left of a solid boundary at . The overlap, the range shaded by both inequalities, is ; the solid edge counts because allows equality. For the greatest integer, take the farthest-right whole number the overlap includes. So the greatest integer satisfying both inequalities is . Grid in 7.