Too Big or Not Too Big, That Is the Question

Learning Focus

Graph and interpret systems of linear inequalities to find all solutions that satisfy several limits at once.

Big idea

A system of inequalities is satisfied by every point that makes all inequalities true. The solution set is the overlapping shaded region.

• Boundary line: dashed for $<,>$; solid for $\le,\ge$.
• Test a point to decide which side to shade.
• A point is a solution only if it works in every inequality.

Worked example

Check $(1,1)$ in the system $y\ge 2x-1$ and $y<-x+6$.

• $1\ge 2(1)-1=1$ is true.
• $1<-1+6=5$ is true.

Both are true, so $(1,1)$ is in the solution region. For a limit like $x+y\le 12$, a total of $5$ brownies leaves room for at most $7$ cookies.

Takeaways

• Solid boundary for $\le,\ge$; dashed for $<,>$.
• The solution is the overlap of all shaded regions.
• Test a point in each inequality.