System of Linear Inequalities

This is a graph of a linear inequality:

Graph of y <= x + 2 with shaded region below the solid line
The inequality y ≤ x + 2

We can see the y = x + 2 line, and the shaded area is where y is less than or equal to x + 2

A Linear Inequality is like a Linear Equation (such as y = 2x+1) ...

... but it will have an Inequality like <, >, ≤, or ≥ instead of an =.

We can have more than one inequality, which makes it a system of inequalities.

Graph of a system of inequalities showing overlapping shaded regions
A System of Linear Inequalities

The Inequality Grapher handles them beautifully, give it a try.

Is a Point Within the Inequalities?

We simply check that the point satisfies each inequality.

Graph of a system of inequalities showing overlapping shaded regions

Check: y ≤ x + 2:

Try (3,1):

1 ≤ 3 + 2

1 ≤ 5

That's True!

Let's check the other inequality y ≥ −x/2:

Try (3,1):

1 ≥ −3/2

1 ≥ −1.5

That's also True!

Both inequalities are true, so the point is within the system of linear inequalities.