# Systems of Linear Inequalities

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We can solve systems of linear inequalities either algebraically or graphically.

## Solving Systems of Linear Inequalities Algebraically

When solving systems of linear inequalities (or simultaneous inequalities):
Step 1: Solve each inequality separately
Step 2: Find the common values between the two inequalities. (Use a number line if necessary)

Example:

Solve the following simultaneous inequalities and represent your solution set on the number line.
x + 2 < 6 and x – 3 > – 1

Solution:
x + 2 < 6
x < 6 – 2
x < 4

x – 3 > – 1
x > – 1 + 3
x > 2

Common values: 2 < x < 4

Example:

Find the integer values of x satisfying the simultaneous inequality:
6 < 2 – 3x < 14

Solution:
Write 6 < 2 – 3x < 14 as two separate inequalities and solve them separately.
6 < 2 – 3x
3x < 2 – 6
3x < – 4
x <

2 – 3x < 14
– 3x < 14 – 2
– 3x < 12
3x > –12
x > – 4
Common values: – 4 < x <

The integer values of x are  –3 and –2

## Solving System of Linear Inequalities Graphically

We can also solve systems of linear inequalities graphically. The following videos show some examples of solving systems of linear inequalities graphically

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