40.4 Solution Half-Plane Selection

Solution Half-Plane Selection is the process of determining which of the two regions created by a boundary line satisfies a given linear inequality, using a test point substituted directly into the inequality to decide which side of the boundary should be shaded.

Boundary-Free Test Point Selection

Procedure

A point that does not lie on the boundary line itself is chosen from the coordinate plane, since a point on the boundary cannot indicate which side of the line satisfies the inequality.

Example

For the boundary line $y=2x+1$, a point such as $(0,0)$ is selected, provided it is confirmed not to lie on this line.

Coordinate Substitution into the Inequality

Procedure

The horizontal and vertical coordinates of the chosen test point are substituted directly into the original inequality, in place of the corresponding variables.

Example

Substituting $(0,0)$ into $y<2x+1$ gives:

True Test-Point Result

Meaning

If the substitution produces a true numerical statement, the chosen test point lies within the solution region of the inequality.

Example

Continuing the previous substitution, $0<1$ is true, so $(0,0)$ lies within the solution region.

False Test-Point Result

Meaning

If the substitution produces a false numerical statement, the chosen test point lies outside the solution region, on the opposite side of the boundary from where the true solution region lies.

Example

For the inequality $y>2x+1$, substituting the same point gives $0>1$, which is false, so this point does not lie within the solution region.

Test-Point Side Shading

Procedure

When the test point produces a true result, the entire half-plane containing that test point is shaded, representing the full solution region of the inequality.

Opposite Side Exclusion

Procedure

When the test point produces a false result, the half-plane containing that test point is left unshaded, and the opposite half-plane, on the other side of the boundary, is shaded instead.

Reasoning

Since every point on one side of a boundary line produces the same true-or-false outcome when substituted, a single false result at the test point confirms that the entire side containing it must be excluded.

Coordinate Origin as a Test Point

Common Choice

The origin, $(0,0)$, is frequently chosen as the test point, since substituting zero for both variables typically simplifies the arithmetic significantly.

Efficiency

Using the origin whenever possible reduces the substitution to evaluating only the constant terms of the inequality, minimizing the chance of an arithmetic error.

Origin-on-Boundary Limitation

Limitation

The origin cannot be used as a test point whenever the boundary line itself passes through the origin, since a point exactly on the boundary provides no information about which side satisfies the inequality.

Correction

In this situation, a different point not lying on the boundary, such as $(1,0)$ or $(0,1)$, is chosen instead, after confirming it also does not lie on the boundary.

Above-Boundary Inequality Region

Description

When the isolated inequality uses a greater-than or greater-than-or-equal symbol with the vertical variable on the left, such as $y>mx+b$, the solution region lies above the boundary line.

Below-Boundary Inequality Region

Description

When the isolated inequality uses a less-than or less-than-or-equal symbol with the vertical variable on the left, such as $y<mx+b$, the solution region lies below the boundary line.

Left-of-Boundary Inequality Region

Description

For a vertical boundary line, an inequality such as $x<a$ places the solution region to the left of the boundary line, containing all points with a horizontal coordinate smaller than $a$.

Right-of-Boundary Inequality Region

Description

For a vertical boundary line, an inequality such as $x>a$ places the solution region to the right of the boundary line, containing all points with a horizontal coordinate larger than $a$.

Consistency with the Test-Point Method

Regardless of these directional shortcuts, substituting a test point remains the most reliable way to confirm which side of any boundary, whether slanted, vertical, or horizontal, contains the true solution region.