# How To Solve Multi Step Inequalities

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Definition: - Inequalities which includes more than one inequality symbols is known as multi step inequalities.

Examples of multi step inequalities: -
·        2 x + 1 < 5 x < 6 x + 9

·        5 > 3 x + 1 > 2 x – 2
·      -3 ≤ (9 x – 1) / (2 x + 3) / 9
·       30 x ≥ (x + 1) / 2 ≥ 0

Example 1: - Solve the following multi step inequalities and find x.
2 x + 1 < 5 x < 9 – 6 x

Solution: - Solve the two inequalities separately
i)                 2 x + 1 < 5 x
(2 x + 1) – 1 < 5 x – 1
2 x < 5 x – 1
2 x – 5 x < (5 x – 1) – 5 x
-3 x < - 1
- 3 x / - 3 > - 1 / - 3           (If we multiply both sides by a negative number then the sign will flip.)
x > 1 / 3

ii)               5 x < 9 – 6 x
5 x + 6 x < (9 – 6 x) + 6 x
13 x < 9
13 x / 13 < 9 / 13
x < 9 / 13
Therefore 1/3

Question 2: - 5 > 3 x + 1 > 4x + 1

Solution: -
i)             5>3x+1
5-1 > 3x+1-1
4 > 3x
4/3 > x

ii)            3x+1 > 4 x + 1
3x+1-4x > 4x + 1 -4x
-x > 1
X < -1