Reason inside simultaneous constraints

A feasible choice must satisfy every restriction at once. An inequality boundary is only one part of the region, and integer requirements can change the final answer.

Choose a representation

At a fixed input, translate each inequality into an allowed output interval. Intersect the intervals, then apply whole-number restrictions or evaluate an objective at its feasible extreme.

Worked example

Nonnegative numbers x,y satisfy 2 x+3 y≤26. If x=4, what is the greatest possible y? Substitute the fixed x into the full resource constraint. Subtract its contribution, then divide the remaining capacity by 3. Substitute the given numbers: (26-2 × 4) ÷ 3 = 6. The requested value is 6.

Contrast a wrong operation

In that example, 18 follows this error: Report unused capacity without converting it to y units. The correct requested value is 6. Explain which quantity the mistaken method is actually calculating.

Connect a second form

How many integers x satisfy |x−6|<4? The strict distance restriction is 6−4<x<6+4. Exclude both integer endpoints and count the integers inside. Substitute the given numbers: 2 × 4-1 = 7. The requested value is 7.

Transfer and check

Nonnegative numbers x,y satisfy 2 x+3 y≤31. If x=5, what is the greatest possible y? Pause before checking. Answer: 7. Substitute the fixed x into the full resource constraint. Subtract its contribution, then divide the remaining capacity by 3. Substitute the given numbers: (31-2 × 5) ÷ 3 = 7.

Boundaries of the method

Meeting one constraint does not guarantee meeting another. A strict inequality excludes its boundary even when the algebra produces an integer. This lesson samples a PSAT/NMSQT objective; its examples are original and do not claim official question status or full coverage.

WORKED EXAMPLE

The shaded region shows the allowed nonnegative values of x and y. When x=4, what is the greatest possible y?

The feasible region is x≥0, y≥0,2x+3y≤26. A vertical guide marks x=4.

Substitute the fixed x into the full resource constraint.
Subtract its contribution, then divide the remaining capacity by 3.
Substitute the given numbers: (26-2 × 4) ÷ 3 = 6.
The requested value is 6.

Answer: 6

Compare the other answer choices

Report unused capacity without converting it to y units.

22/3

Subtract x instead of twice x.

26/3

Ignore the capacity used by x.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

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