ALGEBRA INEQUALITIES - INEQUALITY, INEQUATION, DOMAIN, SOLUTION , EXAMPLES, LINKS

Please study Math Equations before Algebra Inequalities,
if you have not already done so.

The definitions given there, such as Mathematical Sentence,
Mathematical Statement, Open Sentence are relevant here also.

There, we also discussed the solution set or roots
and Domain of the variable with respect to Equations.

Now we define them with respect to Inequations.

Inequalities : Algebra Inequalities

We have seen, A true mathematical statement containing
the sign ' is equal to' ( = ) is called an Equality.

Similarly, a true statement containing the signs 'greater than' (>) or 'less than' (<) is called an inequality.

Observe that inequality sign is always pointed towards the lesser number.

Sometimes inequality may also contain

• ≠ read as not equal to.
• ≥ read as greater than or equal to.
• ≤ read as less than or equal to.

• a ≠ b means a > b or a < b
• a ≥ b means a > b or a = b which means a is not less than b.
• a ≤ b means a < b or a = b which means a is not greater than b.

Some times two or more inequalities are written in one sentence.
e.g, 3 < x < 9 means x > 3 and x < 9 i.e. x lies between 3 and 9.

You may note that

x > 0 means x is a positive number and x < 0 means x is a negative number.

Inequation : Algebra Inequalities

An open sentence containing one of the signs ≠, <, ≤, >, or ≥ is called an inequation.

As in equations, here also the variable
is denoted by letters of the alphabet.

An inequation is also called Algebra Inequality.

Solution Set or Roots of an Inequation : Algebra Inequalities

Just like in equations, a replacement number for the variable which make the inequation true is called a Solution or Root of the Inequation. The set of solutions of an inequation is called the Solution Set.

Solving an inequation means finding all the roots of the inequation.

Domain of the variable : Algebra Inequalities

We have seen the replacement value for the variable which
satifies the inequation is the solution for the inequation.

The set of values from which we can replace the variable
is called Replacement set.

The replacement set of the variable of an inequation is called the Domain of the variable.

Thus, Solution or root (or) Solution set or roots,
Domain of the variable, carry the same meaning
in inequations as in equations.

Just like in equations, the Domain of the variable in the
inequation has to be specified before trying to solve it.

If no Domain is specified, we take the
Real number set as the Domain.

Let us see a few examples.

Example 1 of Algebra Inequalities

Solve x + 5 > 10. Given that Domain of x is set N (Natural Number set).

We can see for x + 5 to be greater than 10,
the replacement value of x has to be greater than 5.
∴ the solution is x > 5.
As the Domain of x is set N (Natural Number set),
the solution set is x = { 6, 7, 8, 9, ............}.

Example 2 of Algebra Inequalities

Solve x - 3 ≤ 0, given that Domain of x is the set { 3, 4, 5, 6 }.

When we substitute the values from the Domain set, in place
of x, we can observe that only 3 satifies the given Inequality.

Thus x = 3 is the solution.

Example 3 of Algebra Inequalities

Solve 10 - x ≥ 20, where Domain of x is the Whole Number set, W.

We know, there is no whole number, which when
subtracted from 10 gives a value more than 20.

Hence, there is no solution or root for
the inequation in the given domain.

Now, observe the inequations and their
solutions in the three examples given above.

In Example 1, the inequation has more than one root.
In Example 2, the inequation has one root.
In Example 3, the inequation has no root at all.

So we can say,

An inequation may have a single root, more than one root or no root at all.

For further discussion and solution of more problems, go to

Linear Inequalities

and

Quadratic Inequalities.