# Inequality

Inequality is when one object is:

• smaller than the other (${\displaystyle \ a means that a is smaller than b)
• bigger than the other (${\displaystyle \ a>b}$ means that a is bigger than b)
• not smaller than the other (${\displaystyle a\geq b}$ means that a is not smaller than b, that is, it is either bigger, or equal to b)
• not bigger than the other (${\displaystyle a\leq b}$ means that a is not bigger than b, or it is smaller or equal to b)

Inequality is sometimes used to name a statement that one expression is smaller, greater, not smaller or not greater than the other.

## Working with inequalities

Inequality 1 This is the solution for the equation x+4>12

Inequality in math is when two solutions or answers are compared by greater than or less than. It is when the two or yet many solutions are being compared is not of equal amount. Solving an inequality means finding its solutions. When you substitute a number to a variable and the statement is true, then it is a solution. When you substitute a number to a variable and the statement is not true then, the number is not a solution to the statement.

Inequality is finding a solution to a given variable. It is finding a relative order of a set. Inequality have many solutions but you need to find the real solutions. Inequality is solving real numbers. The proper way to read inequality is from left to right, just like the other equations, but the only difference is they have different rules for every equation.

For example, x+4>12, where x is a real number. First, a person needs to find the x and he/she need to know if it is a solution. The answer will be x>8 and it is a true statement. This expression is about the location of x within the set of real numbers. A number line is one way to show the location relative to all other real numbers.[1](See figure Inequality 1)

## Different kinds of inequalities

Linear Inequality Example of linear inequality

There are five different kinds of inequalities:

1. The first one is linear inequalities which are an inequality that differentiates the expressions by either less than or equal to, less than or greater than or equal to, greater than. It is one that if we replace the inequality for the equals relation, then the outcome will be a linear equation.
2. The second one is the combinations of inequalities which are to satisfy the inequalities, you must have a number in the solution sets so that the numbers satisfy the inequalities are going to be the values in the crossing of the two solution sets.[2]
3. The third one is inequalities involving absolute values which means that the values can be rephrased as combinations of inequalities that will involve absolute values.
4. The fourth one is called polynomial inequalities means that it is continuous, it means that their graphs do not have any jumps or breaks.
5. Last but not the least, is the rational inequalities, which means that it is the form of one of the polynomial divided by a polynomial. In other words, the rational function graphs do not have any breaks nor represent at the zeros of the denominator.
absolute value Example that shows absolute value

## Four ways to solve Inequalities

There are four ways to solve quadratic equations:

1. Rule number one is you have to add or subtract the same number on both sides.
2. Rule number two is that you have to shift the sides and change the positioning of the sign of the inequality.
3. Rule number three is you have to multiply.
4. Rule number four is to divide the same positive or negative number into the both sides. But, you can only use these on easy inequality problems.[3]
example of multiplication of Inequality

Furthermore, it will take two steps to solve an inequality. The first one is to simplify using the reciprocal of addition or subtraction. The second one is to simplify more by using the reciprocal of multiplication or division. When you are multiplying or dividing an inequality by a negative number, remember to turn the inequality symbol.[4]

## Examples of how to solve Inequalities

Inequality 2 Solution for the equation -6y<-12

Inequality is a mathematical statement that explains that the two values are not equal and different. The equation ab means a is not equal to b. Inequality is the same with any equation but the only difference is that inequality does not use an equal sign instead it uses symbols. The inequality b>a represents that b is greater than a. Speed limits,mark, and others use inequality to express them.

When solving an inequality a person need to have a true statement. When you divide or multiply an inequality with a negative number on both sides the statement is false.In order to make the statement correct with a negative number, you need to reverse the symbol to make that statement correct. When a number is a positive number you don’t need to reverse the symbol. Inequality is about making a true statement.

For example, start with a true statement -6y<-12. When both sides are divided by -6 the result will become y<2. In this statement the symbol need to be reversed in order to have a true statement, y>2 is the correct answer. In the number line (see figure Inequality 2), a closed shaded circle points out that it is included in the solution set. An open circle points out that it is not included in the solution set.[5]

## References

1. "Inequality". Maddocks, J.R. "Inequality." The Gale Encyclopedia of Science. Ed. K. Lee Lerner and Brenda Wilmoth Lerner. 4th ed. Vol. 3. Detroit: Gale, 2008. 2279-2281. Gale Virtual Reference Library. Web. 22 May 2016..
2. "Linear equations". Khamsi, Mohamed Amine. "Linear Equations". Sosmath.com. N.p., 2016. Web. 23 May 2016..
3. "Equations and Inequalities". "Equations And Inequalities - Two-Step Equations And Inequalities - First Glance".Math.com. N. p., 2016. Web. 26 May 2016..
4. "Inequalities". Michael, Michelle R. "Inequalities." Mathematics. Ed. Barry Max Brandenberger, Jr. Vol. 2. New York: Macmillan Reference USA, 2002. 131-133. Gale Virtual Reference Library. Web. 22 May 2016..