# How To Solve Compound Inequalities

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We have learned about the simple linear inequality like an x + b > c but sometimes the inequality may be like      d < an x + b < d.

Examples of compound inequalities: -
·          1 < 2 x + 3 < 3
·         -2 ≥ 4 x + 5 ≥ 5
·          x + 3 > 4 x > 5 x – 5
·        -2 ≤ 5 x + 9 ≤ 9

How to solve compound inequalities: -

Question 1: - If 1 < 2 x + 3 < 3, then find x.

Solution: -
i)  Separate the inequality like
1 < 2 x + 3                        and     2 x + 3 < 3

ii)  Solve each of these inequalities separately like a simple linear inequality.
1 < 2 x + 3
1 – 3 < 2 x + 3 – 3
2 < 2 x
2 / 2 < 2 x / 2
1 < x

And 2 x + 3 < 3
2 x + 3 – 3 < 3 – 3
2 x < 0
2 x / 2 < 0 / 2
x < 0

Therefore, –1< x<0.

Question 2: - If x + 3 > 4 x > 5 x - 5, find x.

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

ii)   4 x > 5 x - 5
4 x – 5 x > 5 x - 5 – 5x
x  > -5
x < 5

Therefore, 1