Math 1 · Unit 5 · Lesson 4 · part 2 of 6
A limit becomes a region
A real limit is called a constraint : a rule the answer must obey.
A limit becomes a region
A real limit is called a constraint: a rule the answer must obey. Spend at most $20. Fit in a van that seats 7. Use no more than 12 feet of fencing. Every constraint like these is an inequality, and its graph is every combination that obeys the rule.
The story
Hot dogs cost $2 and drinks cost $3. You have $20 at most. Let x be the number of hot dogs and y the number of drinks.
The cost is 2x + 3y, and it must be at most 20: 2x + 3y ≤ 20.
Graph it (Lesson 2)
Solve for y: 3y ≤ −2x + 20, so y ≤ −23x + 203. Boundary from m and b, solid (or-equal), shade below. Test (0, 0): 0 ≤ 20, true, so (0, 0)'s side. Also x ≥ 0 and y ≥ 0, since you cannot buy a negative hot dog, so the region stays in the corner.
Read it
Every point in the shaded region is a combination you can afford: feasible, allowed. The boundary line is spending exactly $20, the most you can buy. Inside is under budget. Outside is over.