Inequalities in Maths

Inequalities characterise the connection between two values that are not equal in mathematics. Inequality refers to a lack of equality. When two values are not equal, we use the “not equal sign ()” in most cases. However, various inequalities are employed to compare the numbers, whether they are less than or higher than. In this post, we’ll look at the many types of inequalities used in algebra, as well as inequality symbols, properties, and how to solve linear inequalities in one and two variables.

Equations in mathematics aren’t usually about balancing both sides with a ‘equal to’ sign. It’s sometimes about a ‘not equal to’ connection, when something is larger than or less than the other. In mathematics, an inequality is a relationship that compares two numbers or other mathematical expressions in a non-equal way. Inequalities are mathematical expressions that fall within the category of algebra.

Inequality Definition

Inequalities are mathematical expressions with unequal sides on both sides. Between the equal signs, a less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign is used to substitute the equal sign.

• P ≠ q indicates that the value of p is not equal to the value of q

• P < q indicates that q is greater than p

• P > q indicates that p is greater than q

• P ≤ q indicates that p is less than or equal to q

• P ≥ q indicates that p is greater than or equal to q.    

Inequalities come in a variety of forms. The following are some of the most significant inequalities:

• Polynomial inequalities

• Absolute value inequalities

• Rational inequalities

Basics of Inequalities 

Inequalities have their own set of rules. Here are a few with instances of inequality:

Inequalities Rule 1

You can leap over the intermediate inequality when disparities are connected together.

• If the inequalities p < q and q < d hold true then p < d is true

• If the inequalities p > q and q > d hold true then p > d is true

Example: If Oggy is older than Mia and Mia is older than Cherry, then Oggy must be the older of the two.

Inequalities Rule 2

When we swaps two number then this holds :

• If the inequalities p > q holds true then we can write it as q < p

• If the inequalities p < q holds true then we can write it as q < p

For instance, because Oggy is older than Mia, Mia is younger than Oggy.

Inequalities Rule 3 

Inequalities is to add the number d to both sides of the inequality.

 If p < q, 

Then p + d < q + d

Oggy, for example, has less money than Mia. Even if Oggy and Mia each receive an additional $5, Oggy will still have less money than Mia.

Likewise:

If p < q, then p – d < q – d

If p > q, then p + d > q + d, and

If p > q, then p – d > q – d

As a result, adding and subtracting the same value from both p and q has no effect on the inequality.

Inequalities Rule 4

If we multiply two numbers p and q by a same positive number, there will be no change in the sign of inequality. If you multiply both p and q by a negative number, the inequality swaps: p<q becomes q<p after multiplying by (-2)

Here are the rules

• If p < q, and d is positive, then pd < qd

• If p < q, and d is negative, then pd > qd (inequality swaps)

Example of a positive case: Oggy’s score of 5 is less than Mia’s score of 9 (p q). Even if Oggy and Mia double their scores, 2p<2q, Oggy’s score will be lower than Mia’s. If the scores go negative, the scores will be -p > -q.

Inequalities Rule 5

The direction of the inequality is changed by putting minuses in front of p and q.

• If p < q then −p > −q

• If p > q, then −p < −q

It’s the same as multiplying by (-1) only it’s in the opposite way.

Inequalities Rule 6

Taking the reciprocal 1/value of both p and q reverses the inequality’s direction. When both a and b are positive or negative at the same time then

• If, p < q, then 1/p > 1/q

• If p > q, then 1/p < 1/q

Inequalities Rule 7

A number’s square is always bigger than or equal to zero, hence p² ≥ 0.

For instance, (4)² = 16, (−4)² = 16, (0)² = 0

Inequalities Rule 8:

The inequity will not be changed by taking the square root. If p≤ q, then √p ≤ √q (for p, q ≥ 0).

For example, if p=2 and q=7, 

2 ≤ 7, then √2 ≤ √7

Conclusion

We never get a closed interval in the solution if the symbol is strictly less than or strictly bigger than. Because they are not integers to include, we always receive ∞ or -∞. When calculating rational inequalities, always write open intervals at excluded values. Only in the event of justified disparities should excluded values be considered.