The property of real numbers which guarantees that for any two real numbers a and b, exactly one of the following must be true.
Either a<b, a=b or a>b.
Such property of whole numbers is known as Trichotomy property. The sign (=) 'equal to' (<) 'lesser than' and (>) 'greater than are the trichotomy sign.
Negation of trichotomy
Suppose any two numbers 3 and 5.
Here, 3<5 or 5>3 are a true comparison. But 3\(\nless\)5 (3 is not less than 5) or 5\(\ngtr\)3 (5 is not greater than) are the false comparison.
Here, 3\(\nless\)5 is the negation of 3<5 and 5\(\ngtr\)3 is the negation of 5>3.
Thus '\(\nless\)' (is not less than) is the negation of '<' (is less than) '\(\ngtr\)' (is not greater than) and '≠' (is not equal to) is the negation of '=' (equal to).
Trichotomy Rules
Inequalities
When an open statement contains the signs of less than '<', greater than '>', less than equal to (≤) or greater than equal to (≥) it is known as an inequation. An inequation is also called inequality.
x>10, y<4, p≥-5 etc are the examples of inequalities.
Replacement set and solution set
The set of values of x that makes the inequality true is {1,2,3} then it is known as the solution set. However, the set of natural numbers, N = {1,2,3,4,5} from which numbers are used to replace x in the inequality is known as the replacement set. For examples,
N = {1,2,3,4,5}
When x = 1, 1<4 (It is true)
When x = 2, 2<4 (It is true)
When x = 3, 3<4 (It is true)
When x = 4, 4<4 (It is false)
When x = 5, 5<4, (It is false)
Graphical representation of solution set
For the graphical representation of solution sets, we use number lines to show the solution sets of the given inequalities.
Trichotomy Rules
Solution:
Here, the replacement set, R = {1, 2, 3, 4, 5}
The given inequality is,
or, 2x + 3 < 11
or, 2x + 3 - 3 < 11 - 3
or, 2x < 8
or, \(\frac{2x}{2}\) < \(\frac{8}{2}\)
or, x < 4
Now,
when x = 1, 1 < 4 which is true
when x = 2, 2 < 4 which is true
when x = 3, 3 < 4 which is true
when x = 4, 4 < 4 which is true
∴ Solution set = {1, 2, 3}
Solution:
or, 3x - 2 > x + 10
or, 3x - 2 - x > 10
or, 2x - 2 > 10
or, 2x - 2 + 2 > 10 + 2
or, 2x > 12
or, \(\frac{2x}{2}\) > \(\frac{12}{2}\)
or, x > 6
∴ Solution set = {7, 8, 9, . . . .. . . }
Solution:
Let the number be x
According to the given statement,
3x - 7 ≤ -10
or, 3x - 7 + 7 ≤ -10 + 7 (7 is added to both sides.)
or, 3x ≤ -3
or, \(\frac{3x}{3}\) ≤ \(\frac{-3}{3}\) (Both sides are divided by 3)
or, x ≤ -1
∴ Solution set = {-1, -2, -3, . . . . . }
Solution:
Let the number be x.
According to the given statement.
3 - \(\frac{x}{4}\) ≥ 2
or, 3 - 3 - \(\frac{x}{4}\) ≥ 2 - 3 (3 is subtracted from both sides)
or, -\(\frac{x}{4}\) ≥ - 1
or, -4 × -(\(\frac{x}{4}\)) ≤ -1 × (-4) (Both sides are multiplied by -4. So, the sign ≥ is change into ≤ .)
or, x ≤ 4
∴ Solution set = {4, 3, 2, 1, 0, -1, . . . . .. . . }
The property of whole numbers is known as ______ property.
An inequation is also called ______ .
______ is an open statement containing 'equal to' (=) sign.
If the open statement contains the sign of 'less than' (<), 'greater than' (>), 'less than and equal to' (≤), or 'greater than and equal to' (≥), it is known as ______ .
ASK ANY QUESTION ON Trichotomy and Inequalities
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