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Grade 7 - Mathematics
6.33 Properties of Inequalities - IV

Examples:
  1. Consider the numbers -2, 3 and 5.
    We observe that -2 < 3, 3 < 5 and -2 < 5
    \ If -2 < 3 and 3 < 5 then -2 < 5.
  2. Consider the numbers -6, -3 and 6.
    We observe that 6 > -3, -3 > -6 and 6 > -6.
    \ If 6 > -3 and -3 > -6 then 6 > -6.

If three numbers are related in such a way that the first is less (greater) than the second and the second is less (greater) than the third, then the first is less (greater) than the third.
This is called the transitive property.
Double Inequality:
Example: Solve the double inequality
-3 < 4(x + 2) - 3 < 9
Solution:
-3 < 4(x + 2) - 3 < 9
Using distributive law 4(x + 2) can be written as 4x + 8
-3 < 4x + 8 - 3 < 9
-3 < 4x + 5 < 9
Subtracting 5 to all three terms and simplifying we get:
-3 - 5 < 4x + 5 - 5 < 9 - 5
-8 < 4x < 4
Dividing all three terms by 4 we get:
-2 < x < 1
x >-2 and x <1 implies that x ranges from -2 to 1.


Directions: Solve for the variable in the inequalities given below. Write at least ten examples of your own and prove transitive property.
Q 1: (-1) < -2(X-3)- 3 < 7
2 < x < -2
2 > x > -2

Q 2: (-2) < -1(2x-4) < 8
3 < x < -2
3 > x > -2

Q 3: 8 < 2(a+2) < -2
2 > a > -3
2 < a < -3

Q 4: 9 < 4x-3 < 17
3 > x > 5
3 < x < 5

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