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Grade 9 · lesson 2 of 4

Test a Point against Every Constraint

A feasible point satisfies all listed restrictions at once.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Which point satisfies x≥0, y≥0 and x+y≤4? Candidate answers: (A) (2,2) (B) (2,−1) (C) (3,2) (D) (−1,2).
  • Independent check inputs: Which point satisfies x≥0, y≥0 and x+y≤6? Candidate answers: (A) (3,3) (B) (3,−1) (C) (4,3) (D) (−1,3).

Learn the idea

A mathematical plan can require simultaneous equalities or inequalities. The feasible set contains inputs satisfying all constraints, and a linear objective asks which allowed choice gives the greatest or least modeled value. A feasible point satisfies all listed restrictions at once. Nonnegative coordinates and an inclusive sum cap can be checked directly without drawing the entire feasible region.

Here is the method on the supplied teaching case. The point (2,2) has nonnegative coordinates and sum 4, equal to the inclusive cap. The first distractor exceeds the cap; the others use a negative coordinate.

Passing one inequality does not make a point feasible. Test each constraint separately, retaining equality for an at-most cap; sign restrictions often rule out otherwise plausible points.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Which point satisfies x≥0, y≥0 and x+y≤4? Candidate answers: (A) (2,2) (B) (2,−1) (C) (3,2) (D) (−1,2). Solution: The point (2,2) has nonnegative coordinates and sum 4, equal to the inclusive cap. The first distractor exceeds the cap; the others use a negative coordinate. The final answer is (2,2). Passing one inequality does not make a point feasible. Test each constraint separately, retaining equality for an at-most cap; sign restrictions often rule out otherwise plausible points.

Your turn

Check your reasoning

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Helpful tips
  • A feasible point satisfies all listed restrictions at once.
  • Passing one inequality does not make a point feasible.
Which point satisfies x≥0, y≥0 and x+y≤6?

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

(3,3)

The point (3,3) has nonnegative coordinates and sum 6, equal to the inclusive cap. The first distractor exceeds the cap; the others use a negative coordinate.

Take it further

Changed-input task: Which point satisfies x≥0, y≥0 and x+y≤18? Candidate answers: (A) (10,9) (B) (9,9) (C) (9,−1) (D) (−1,9). Show the mathematical reason, not only the selected answer. Expected result: (9,9). The point (9,9) has nonnegative coordinates and sum 18, equal to the inclusive cap. The first distractor exceeds the cap; the others use a negative coordinate.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Which point satisfies x≥0, y≥0 and x+y≤8?
Need a hint?
  • A feasible point satisfies all listed restrictions at once.
  • Passing one inequality does not make a point feasible.
2. Which point satisfies x≥0, y≥0 and x+y≤10?
Need a hint?
  • A feasible point satisfies all listed restrictions at once.
  • Passing one inequality does not make a point feasible.
3. Which point satisfies x≥0, y≥0 and x+y≤12?
Need a hint?
  • A feasible point satisfies all listed restrictions at once.
  • Passing one inequality does not make a point feasible.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.