40.3 Inequality Boundary Graphing

Inequality Boundary Graphing is the process of drawing the line associated with a linear inequality onto a coordinate plane, choosing between a solid or dashed style based on whether the boundary itself is included in the solution, in preparation for shading the appropriate region.

Boundary Equation Formation

Procedure

Every linear inequality has an associated boundary equation, formed by taking the same expression on both sides of the inequality and connecting them with an equals sign instead of an inequality symbol.

Example

The inequality $2x+y\le 6$ has the boundary equation:

Inequality Symbol Replacement by Equality

Purpose of This Step

Replacing the inequality symbol with an equals sign converts the inequality into a standard linear equation, allowing all previously established methods for graphing lines to be applied directly.

Result

This boundary equation graphs as a single straight line that separates the coordinate plane into the region satisfying the inequality and the region that does not.

Strict Inequality Dashed Boundary

Rule

When the original inequality uses a strict symbol, either greater than or less than, without an "or equal to" component, the boundary line is drawn as a dashed line.

Meaning

A dashed boundary indicates that points lying exactly on this line do not satisfy the original inequality and are excluded from the solution set.

Inclusive Inequality Solid Boundary

Rule

When the original inequality uses an inclusive symbol, either greater than or equal to, or less than or equal to, the boundary line is drawn as a solid line.

Meaning

A solid boundary indicates that points lying exactly on this line do satisfy the original inequality and are included in the solution set.

Slope-Intercept Boundary Construction

Procedure

When the boundary equation is in, or converted into, slope-intercept form, the line is drawn by plotting the vertical intercept and then using the slope to locate a second point.

Example

For the boundary equation $y=-2x+6$, the intercept $(0,6)$ is plotted first, followed by a second point located using the slope of $-2$.

Two-Point Boundary Construction

Procedure

When the boundary equation is in standard form, two convenient points, often the horizontal and vertical intercepts, are found by substitution and used to draw the line directly.

Example

For $2x+y=6$, setting $x=0$ gives the point $(0,6)$, and setting $y=0$ gives the point $(3,0)$.

Vertical Boundary Line Construction

Procedure

When the inequality involves only the horizontal variable, its boundary is a vertical line, constructed by locating the fixed horizontal coordinate and drawing a line straight up and down through it.

Example

For $x>2$, the boundary line is the vertical line $x=2$, drawn as dashed since the inequality is strict.

Horizontal Boundary Line Construction

Procedure

When the inequality involves only the vertical variable, its boundary is a horizontal line, constructed by locating the fixed vertical coordinate and drawing a line straight across through it.

Example

For $y\le 3$, the boundary line is the horizontal line $y=3$, drawn as solid since the inequality is inclusive.

Boundary Style and Symbol Agreement

Final Check

After drawing the boundary, its style, solid or dashed, is compared directly against the original inequality symbol to confirm they match: strict symbols require dashed lines, and inclusive symbols require solid lines.

Consequence of a Mismatch

A boundary style that does not match its inequality symbol will lead to an incorrect interpretation of whether points on the line belong to the solution set, so this agreement must be confirmed before any shading is applied.