To check a point, substitute its coordinates into every inequality and confirm that each one is true. One false statement is enough to reject the point, and all true statements are needed to accept it.
This is the final skill in the inequalities and feasible regions module. It works with or without a graph, and it is the same test you used to choose shading sides in drawing a boundary.
How do you test a point?
- Write the constraints in a list so none is missed.
- Substitute x and y into each one.
- Write the statement and mark it true or false.
- Pay attention to strict symbols: a value equal to the limit fails < or >.
- Write a conclusion: all true means the point is feasible, and any false means it is not.
Worked example
Constraints: y ≥ 1, x + y ≤ 7 and y < 2x. Test the points A (3, 4), B (2, 4) and C (5, 3).
Point A (3, 4):
- y ≥ 1: 4 ≥ 1, true.
- x + y ≤ 7: 7 ≤ 7, true (equality is allowed).
- y < 2x: 4 < 6, true. All true, so A is feasible.
Point B (2, 4):
- y ≥ 1: 4 ≥ 1, true.
- x + y ≤ 7: 6 ≤ 7, true.
- y < 2x: 4 < 4, false, because 4 is not less than 4. One false, so B is not feasible. It lies on the dashed boundary y = 2x.
Point C (5, 3):
- y ≥ 1: true.
- x + y ≤ 7: 8 ≤ 7, false. So C is not feasible. We can stop at the first false statement, although completing the list is good practice.
The mistake to watch for
Mistaken working: “B is (2, 4). 4 ≥ 1 is true and 2 + 4 = 6 ≤ 7 is true, so B is feasible.”
The student stopped after two tests and skipped the third.
The third constraint failed. The correction is to keep a list of all the constraints and tick each one off. A point is accepted only after the last line of the list.
A word-problem version
A school sells x plain cups and y printed cups. At most 20 cups are sold in total, at least 5 are plain, and the number of printed cups is at least twice the number of plain cups. The constraints are x + y ≤ 20, x ≥ 5 and y ≥ 2x.
Is (5, 10) allowed? 15 ≤ 20 is true, 5 ≥ 5 is true and 10 ≥ 10 is true. Yes.
Is (6, 10) allowed? 16 ≤ 20 and 6 ≥ 5 are true, but 10 ≥ 12 is false. No.
Check yourself
1. Constraints: x + y ≤ 10, x ≥ 2, y > 3. Is P (4, 4) feasible? Is Q (6, 3)? Is R (7, 4)?
Show answer
P: 8 ≤ 10, 4 ≥ 2 and 4 > 3 are all true. Feasible.
Q: y > 3 becomes 3 > 3, which is false. Not feasible.
R: x + y = 11 and 11 ≤ 10 is false. Not feasible.
2. Is (3, 4) on the boundary of x + y ≤ 7? Does it satisfy the inequality?
Show answer
3 + 4 = 7, so it lies exactly on the boundary. With ≤, equality is allowed, so yes, it satisfies it.
3. A trip has x adults and y students. There are at most 30 people, at least 2 adults, and y ≤ 8x. Is (3, 24) allowed? Is (2, 17)?
Show answer
(3, 24): 27 ≤ 30, 3 ≥ 2 and 24 ≤ 24 are all true. Allowed.
(2, 17): 19 ≤ 30 and 2 ≥ 2 are true, but 17 ≤ 16 is false. Not allowed.
Where this leads next
Put the whole module to work in the inequalities practice set. Sequences come next in sequences and pattern rules. The non-calculator working trainer supports the substitution arithmetic.
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