40.2 Individual Inequality Preparation

Individual Inequality Preparation is the process of rearranging a single linear inequality into a standardized form suitable for graphing, isolating one variable, correctly handling the inequality symbol during algebraic operations, and identifying the exact parameters needed to draw its boundary line.

Inequality Variable Arrangement

Procedure

The terms of the inequality are arranged so that all variable terms appear on one side and the constant term appears on the other, matching the general structure used for solving linear equations.

Example

The inequality $4x+2y\le 12-y$ is rearranged by moving the $y$ term to the left side:

Inequality Output Isolation for Graphing

Procedure

The vertical variable is isolated on one side of the inequality, producing a form comparable to slope-intercept form, which directly indicates the boundary line's slope and intercept for graphing.

Example

Continuing from above, isolating $y$ produces:

Additive Inequality Transformation

Procedure

Adding or subtracting the same quantity from both sides of an inequality preserves the direction of the inequality symbol, exactly as it would for an equation.

Example

Given $x+3>7$, subtracting three from both sides preserves the direction of the symbol:

Positive Coefficient Division

Procedure

Dividing both sides of an inequality by a positive number preserves the direction of the inequality symbol.

Example

Given $2x<10$, dividing both sides by two, a positive number, preserves the direction of the symbol:

Negative Coefficient Division and Symbol Reversal

Procedure

Dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol, changing greater-than to less-than or less-than-or-equal to greater-than-or-equal.

Example

Given $-2x<10$, dividing both sides by negative two requires reversing the symbol:

Inequality Boundary Graph Parameter Identification

Procedure

Once the inequality is written with the vertical variable isolated, its slope and vertical intercept are identified directly, exactly as they would be for the boundary equation's corresponding linear equation.

Example

From $y\le -\frac{4}{3}x+4$, the boundary line has slope $-\frac{4}{3}$ and vertical intercept $4$.

Standard Form Boundary Preparation

Procedure

If the inequality is given in standard form rather than with an isolated variable, its associated boundary equation is temporarily written by replacing the inequality symbol with an equals sign, and this boundary equation is prepared using standard slope extraction methods.

Example

The inequality $4x+3y\le 12$ has the associated boundary equation $4x+3y=12$.

Vertical Boundary Inequality Recognition

Procedure

An inequality containing only the horizontal variable, such as $x>3$, is recognized as having a vertical boundary line, since its associated boundary equation is a vertical line.

Horizontal Boundary Inequality Recognition

Procedure

An inequality containing only the vertical variable, such as $y\le 2$, is recognized as having a horizontal boundary line, since its associated boundary equation is a horizontal line.