Systems of Linear Inequalities

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, –2 and –1

Solving System of Linear Inequalities Graphically

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

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