maths > algebra

Arithmetic Properties of Inequality

what you'll learn...

overview

In this lesson properties of inequalities are explained.

•  inequations are statement of comparison between two expressions

•  statement of comparison remains unchanged when the expressions are modified per PEMA / CADI

•  statement of comparison is modified when an in-equation is modified with another equation.

•  statement of comparison remains unchanged as per the transitivity property of comparison

It is very important to go through this once to understand in-equalities in algebra.

inequation means 'not equal'

Inequation means "statement of comparison between two quantities".

Consider the inequation $2+1<11/2-0.25×2$$2 + 1 < 11 / 2 - 0.25 \times 2$.

The left hand side of the inequation is $2+1$$2 + 1$. The right hand side of the inequation is $11/2-0.25×2$$11 / 2 - 0.25 \times 2$.

The inequation $2+1<11/2-0.25×2$$2 + 1 < 11 / 2 - 0.25 \times 2$ states that left hand side is less than the right hand side.

changing inequations

Consider the inequation $2+1<11/2-0.25×2$$2 + 1 < 11 / 2 - 0.25 \times 2$. The left hand side be modified without affecting the statement of inequality. The left hand side is an numerical expression which can be modified as per PEMA precedence / CADI Laws and Properties of Arithmetics.

The same for right hand side, which can be modified without changing the statement of inequality.

Consider the inequation $2+1<11/2-0.25×2$$2 + 1 < 11 / 2 - 0.25 \times 2$.

The left hand side $2+1$$2 + 1$ can be modified into $2+1-1+1$$2 + 1 - 1 + 1$ as per the additive identity and inverse properties. The value of the expression is not changed, so the inequation holds true for $2+1-1+1<11/2-0.25×2$$2 + 1 - 1 + 1 < 11 / 2 - 0.25 \times 2$

Note: The left hand side $2+1$$2 + 1$ cannot be modified into $2+1+2$$2 + 1 + 2$ as this changes the value of the expression. That is, the inequation may not be true with this modified expression.

two or more inequations

Consider two inequations $1+1<4-1$$1 + 1 < 4 - 1$ and $4-1<{2}^{3}-3$$4 - 1 < {2}^{3} - 3$. By transitivity property of comparisons, it is derived that $1+1<{2}^{3}-3$$1 + 1 < {2}^{3} - 3$

inequation and equation

Consider an inequation $1<3-1$$1 < 3 - 1$ and an equation $4=2+2$$4 = 2 + 2$.

An inequation and an equation can be added into a new inequation. $1+4<\left(3-1\right)+\left(2+2\right)$$1 + 4 < \left(3 - 1\right) + \left(2 + 2\right)$

inequation and equation can be multiplied into a new inequation. $1×4<\left(3-1\right)×\left(2+2\right)$$1 \times 4 < \left(3 - 1\right) \times \left(2 + 2\right)$ (Note: The equation evaluates to positive value.)

+ve or -ve equation

Consider
the inequation $1<3-1$$1 < 3 - 1$,
the equation $4=2+2$$4 = 2 + 2$ evaluating to a positive value, and
the equation $-4=-2-2$$- 4 = - 2 - 2$ evaluating to a negative value.

An inequation can be combined with an equation to arrive at an inequation derived from them as per the following.

•  Equation (evaluating to either positive or negative value) can be added or subtracted to an inequation. eg: $1+4<\left(3-1\right)+\left(2+2\right)$$1 + 4 < \left(3 - 1\right) + \left(2 + 2\right)$ and $1-4<\left(3-1\right)-\left(2+2\right)$$1 - 4 < \left(3 - 1\right) - \left(2 + 2\right)$.

•  An equation, evaluating to a positive value, can be multiplied or divided to an inequation. eg: $1×4<\left(3-1\right)×\left(2+2\right)$$1 \times 4 < \left(3 - 1\right) \times \left(2 + 2\right)$

•  An equation, evaluating to a negative value, can be multiplied or divided which changes the comparison from smaller to larger. eg: $1×\left(-4\right)>\left(3-1\right)×\left(-2-2\right)$$1 \times \left(- 4\right) > \left(3 - 1\right) \times \left(- 2 - 2\right)$

summary

Numerical Inequations :

Inequations are a statement of comparison between left hand side and right hand side.

Inequation consists of expressions on the left side and right side. These expressions may be modified as per PEMA precedence / CADI Laws and Properties of Arithmetics.

Two in-equations may be combined as per the transitivity property of the comparisons.

An equation can be combined to an inequation to arrive at an inequation derived from them as per the following.
•  An equation can be added or subtracted to an inequation
•  An equation, evaluating to a positive value, can be multiplied or divided to an inequation retaining the statement of comparison.
•  An equation, evaluating to a negative value, can be multiplied or divided to an inequation reversing the statement of comparison.

Outline

The outline of material to learn "Algebra Foundation" is as follows.

→   Numerical Arithmetics

→   Arithmetic Operations and Precedence

→   Properties of Comparison

→   Properties of Multiplication

→   Properties of Exponents

→   Algebraic Expressions

→   Algebraic Equations

→   Algebraic Identities

→   Algebraic Inequations