Systems of linear inequalities

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

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