40.7 System Point Verification

System Point Verification is the process of confirming whether a specific candidate coordinate pair is a valid solution to a system of linear inequalities, by substituting it into every inequality in the system individually and requiring every one of them to produce a true result.

Candidate Coordinate Pair Selection

Procedure

A specific ordered pair, either proposed as a potential solution or read from a graphed common region, is chosen as the candidate to be tested against the system.

Example

For a system of two inequalities, the ordered pair $(1,1)$ might be selected as a candidate to verify.

First Inequality Candidate Test

Procedure

The candidate's coordinates are substituted into the first inequality of the system, and the resulting numerical statement is evaluated for truth.

Example

For the inequality $y\le x+3$, substituting $(1,1)$ gives $1\le 4$, which is true.

Second Inequality Candidate Test

Procedure

The same candidate's coordinates are substituted into the second inequality of the system, independently of the result obtained from the first test.

Example

For the inequality $y>-x+0.5$, substituting $(1,1)$ gives $1>-0.5$, which is true.

Every Inequality Truth Requirement

Rule

A candidate point is accepted as a valid solution to the system only if every single inequality, without exception, produces a true statement upon substitution.

Consequence of a Single Failure

If even one inequality in the system produces a false statement for the candidate point, the point is rejected entirely as a solution to the system, regardless of how many other inequalities it satisfied.

Interior Solution Point

Description

A point that satisfies every inequality in the system with a strict truth, not lying exactly on any boundary line, is classified as an interior solution point.

Included Boundary Solution Point

Description

A point lying exactly on a solid boundary line, associated with an inclusive inequality, is still classified as a valid solution point, since a solid boundary belongs to its inequality's solution set.

Example

For the inclusive inequality $y\le x+3$, a point satisfying $y=x+3$ exactly still produces a true statement and is accepted.

Excluded Boundary Point

Description

A point lying exactly on a dashed boundary line, associated with a strict inequality, is not classified as a valid solution point, since a dashed boundary is not included in its inequality's solution set.

Example

For the strict inequality $y<x+3$, a point satisfying $y=x+3$ exactly produces a false statement and is rejected.

Outside-System Point

Description

A point that fails at least one inequality in the system, whether by falling on the wrong side of a boundary or on an excluded boundary itself, lies entirely outside the system's solution region.

Identification

Such a point produces at least one false result during substitution, immediately disqualifying it regardless of how the remaining inequalities evaluate.

Graph and Substitution Agreement

Verification Requirement

A candidate point's classification through direct substitution must agree with its apparent position on the graphed common region: points shown within the shaded area should test true for every inequality, and points shown outside should fail at least one.

Resolving Disagreement

If a point appears to lie within the shaded region on the graph but fails a substitution test, or the reverse, the graph's shading, the boundary equations, or the substitution arithmetic should be rechecked, since a correctly constructed graph and correct substitution must always agree.