# Math Assignment Help With Inequalities And Trichotomy

## CHAPTER 11 Inequalities and Trichotomy

**11.1 Introduction: **Inequalities between quantities or numbers or equations is the difference between them in respect of their magnitude or order.

- a < b means a is less than b
- a ≤ b means a is less than or equal to b
- a ≥ b means a is greater than or equal to b etc.
- a > b means a is greater than b etc.

In an inequality, you can add or subtract numbers from each side, with an equation.

You can also multiply or divide by a constant. However, if you multiply or divide by a negative number, the inequality sign is reversed.

#### Example: Solve 6(*x* + 4) < 8*x* + 21

6*x* + 24 < 8*x *+ 21

-2*x* < -3

*x* > 3/2 (note: sign reversed because we divided by -2)

Inequalities can be used to describe the range of values of a variable.

E.g. 3 ≤ *x* < 10, means *x *is greater than or equal to 3 but less than 10.

### 11.2 Properties:

**11.2.1 Trichotomy:** The property states that

For any real numbers, *a* and *b*, only one of the following is true:

a < b

a = b

a > b

In set theory, trichotomy is defined as a property that a binary relation (<) has when all of its members satisfy exactly one of the relations listed above.

Strict inequality is an example of a trichotomous relation in this sense

**11.2.2 Transitivity: **The transitivity of inequalities states that

For any real numbers, a, b and c

If a > b & b > c, then a > c

If a < b & b < c, then a < c

If a > b & b = c, then a > c

If a < b & b = c, then a < c

**11.2.3 Addition and Subtraction**: This property deals with addition and subtraction. It states that:-

For any real numbers, a, b, c:

If a < b, then a + c < b + c and a − c < b − c

If a > b, then a + c > b + c and a − c > b − c

**11.2.4 Multiplication and Division**: This property deals with multiplication and division It states that:-

For any real numbers, a, b, c:

If c is positive and a < b, then ac < bc

If c is negative and a < b, then ac > bc

### 11.3 Rules to Solve Inequalities:

Here are some rules for solving inequalities and finding all of its solutions.

A solution of an inequality is a number which on being substituted for the variable makes the inequality a true statement.

**Example:** Consider the inequality

*x *– 4 > 3

Substitute any value, *x = 9*

The inequality becomes 9-2 > 5.

Thus, *x *= 9 is a solution of the inequality.

Substituting *x = -2*

(-2) – 4 > 3

-6 >3 which is not true.

Thus *x* = -2 is NOT a solution of the inequality

**Rule1**. Add/subtract the same number on both sides.

**Example:** take any inequality

has the same solutions as the inequality

*x* – 6 +6 > 3 +6

*Rule2.* Switching sides and changing the orientation of the inequality sign.

**Example:** The inequality 5-x> 4

has the same solutions as the inequality

4 < 5 - x.

If you can notice the sides have been switched and turned the “>” into “<”

Rule 3. Multiply/divide by the same positive or negative number on both sides.

While multiplying or dividing by a negative number, remember to change the orientation.

**Examples:** Solve 4*x > 6*

Divide both sides by 4

Inequality becomes

*x* > 6/4

*x*>3/2

### 11.3.1 Special Cases - A variable in the denominator

For example consider the inequality

2/(*x -1)* >2

In this case we cannot multiply the right hand side by (*x*-1) because the value of *x* is unknown. Since *x* may be either positive or negative, its difficult to decide the inequality sign.

Step1. Equate the denominator to 0.

*x*-1 = 0

Step2: Assume inequality sign is an = sign

(2/*x -1) *= 2 so *x* = 2

Step3: Plot the points *x* = 1 and *x* = 2 on a number line with an open circle because the original equation included “ <”

Step4: check each region individually by arbitrarily picking a value within each region.

For example, the original inequality holds for *x* < 1, but not for *x* > 2 or 1 < *x* < 2.

### 11.4 Inequalities between means

There are many inequalities between means. For example, for any positive numbers *a*1, *a*2, …, *an* we have HM≤ GM ≤ AM ≤ QM,

### 11.5 Logarithms of inequalities

When manipulating inequalities it is sometimes useful to take the logarithm of both sides of the inequality. To do this the following result is needed.

### 11.6 Power inequalities:

Power inequality means inequalities which contain a^{b} type expressions where *a* and *b* are real positive numbers.

Examples

- If x > 0, then

x^{ x}= (1/e)^{1/e} - If x > 0, then

(x^{x})^{x}> x - If x, y, z > 0, then

(x + y)^{z}+(y + z)^{x}+(x+z)^{y}> 2 - For any real distinct numbers a and b

e^{(a+b)/2}

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