# Systems of Linear Inequalities

## Big Picture

One way to solve systems of linear equalities is to graph the inequalities and see if there are any areas on the graph where the inequalities overlap. The places where the inequalities’ graphs overlap are the solutions to the system.

## Key Terms

Linear Inequality: A linear equation with the $$=$$ sign replaced with inequality signs ($$<$$,$$≤$$, $$>$$, or $$≥$$).

Absolute Value: The absolute value of a number is the distance of that number from $$0$$.

## Graphing Linear Inequaltiies

### In Two Variables

A linear equation in slope-intercept form is $$y = mx + b$$. A linear inequality is closely related to a linear equation, except the = sign is replaced with $$<$$, $$≤$$, $$>$$, or $$≥$$.

To draw a linear inequality,

• Write the equation in slope-intercept form $$y = mx + b$$ for graphing.
• Draw the line. If the inequality sign does not include an equal sign($$<$$ or $$>$$), then draw a dashed line. If the inequality sign includes an equal sign ($$≤$$ or $$≥$$), then draw a solid line.
• The line divides the plane into two halves. Shade the half plane that include the points that are part of the solution.
Examples:
Shade the half plane above the line if the inequality is greater than.

Shade the half plane below the line if the inequality is less than.

Test a point on one side of the line (not on the line) to see if the point makes the inequality true. If it does,shade that side of the line. If not, shade the other side.

### In One Variable

Linear inequalities in one variable can also be graphed on the coordinate plane. The line that gets drawn is a horizontal or vertical line. The graph looks like the solution graphed on the number line but stretched vertically.

Example: $$x > 4$$

# Systems of Linear Inequalities cont.

## Graphing Systems of Inequalities

A system of linear inequalities can be solved by graphing.

• Graph each inequality
• Find the regions of solutions that make all the inequalities true.
• Test a point in the different regions to see if the point makes the linear system of inequalities true.

### Two Linear Inequalities

The solutions of two linear inequalities are unbounded regions, which continue infinitely in at least one direction. Here are two examples:

• The orange region is the solution to $$y < 2x+6$$.
• The blue region is the solution to $$y ≥ 2x-4$$.
• The purple region is the solution to both inequalities.

Test a point on one side of the line (not on the line) to see if the point makes the inequality true. If it does,shade that side of the line. If not, shade the other side.

• The orange region is the solution to $$y > 2x+6$$.
• The blue region is the solution to $$y ≤ 2x-4$$.
• Need to shade both regions!

### More than Two Linear Inequalities

The solutions of two linear inequalities can be unbounded or bounded regions. A bounded region is a finite region with three or more sides.
Example:

$$y > 3x-4$$

$$y < -\frac{9}{4}x + 2$$

$$x ≥ 0$$

$$y ≥ 0$$

### Absolute Values

Absolute value inequalities can be re-written as a system of two inequalities.

Example: $$|x| ≥ 2$$
Rewrite as $$x ≤ -2 \text{ or } x ≥ 2$$

Example: $$|y| < 5$$
Rewrite as $$y > -5$$ and $$y < 5$$