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Trichotomy and Inequalities

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Trichotomy

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

  1. When an equal positive number is added to or subtracted from or multiplied or divided by both sides of trichotomy sign the sign remain the same. For example,
    7>5, then 7+2>5+2
    9>6, then 9-4>6-4
  2. When both sides of trichotomy sign are multiplied or divided by an equal negative number, the sign '<' is changed to '>' and the sign '>' is changed to '<'. For example,
    6>5, then 6×(-2)<5×(-2) (a>b then a(-c))<b×(-c)

Inequalities

source: www.slideshare.net Fig: Inequality
source: www.slideshare.net
Fig: Inequality

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.



  • The property of real numbers which guarantees that for any two real numbers a and b, exactly one of the following must be true.
  • Trichotomy Rules

    1. When an equal positive number is added to or subtracted from or multiplied or divided by both sides of trichotomy sign the sign remain the same.
    2. When both sides of trichotomy sign are multiplied or divided by an equal negative number, the sign '<' is changed to '>' and the sign '>' is changed to '<'. 
  • For the graphical representation of solution sets, we use number lines to show the solution sets of the given inequalities.
.

Very Short Questions

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}

s 

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, . . . .. . . }

s 

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, . . . . . }

s

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, . . . . .. . .    }

s

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  • The property of whole numbers is known as ______ property.

    Trichotomy
    Graphical
    Inequality 
    Negation
  • An inequation is also called ______ .

    division axiom
    trichotomy 
    inequality
    graphic
  • ______ is an open statement containing 'equal to' (=) sign.

    Inequality
    Equation
    Graph
    Trichotomy 
  • 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 ______ .

    trichotomy 
    negation
    inequation 
    graph
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