EDGE Math 8

Solutions to Linear Inequalities in Two Variables

Consider a linear equation in two variables; for instance, $2 x + y = 6.$ If, in this equation, the equality symbol = is replaced by an inequality symbol such as <, $\leq$ , $\geq$ , or >, the resulting statement is called a linear inequality in two variables.

In general, a linear inequality in two variables can be written in any of the following forms:

A x + B y < C,

A x + B y \leq C,

A x + B y > C,

A x + B y \geq C,

where A, B, and C are constants, and A and B are both not equal to zero.

The following are examples of linear inequalities in two variables:

x - 2 y < 4,

y > 5 x,

and

x + y - 2 \leq 0.

Example

Translate the following statement into an inequality in two variables:

The total number of recycled metal cans and plastic bottles is at least 400.

Solution

Let x be the number of recycled metal cans, and y be the number of recycled plastic bottles. Therefore,

x + y \geq 400.