## Inequalities

In mathematics, an inequality is a statement that compares two values, expressions, or quantities using one of the inequality symbols: "\( < \)" (less than), "\( > \)" (greater than), "\( \le \)" (less than or equal to), "\( \ge \)" (greater than or equal to) or "\(\neq \)" (not equal to).

**Comparing numbers and expressions** involves determining the relative sizes of different quantities. There are several methods for comparing numbers and expressions, including:

**Comparison symbols:**One of the simplest methods for comparing numbers is to use comparison symbols such as "less than" \( (<) \), "greater than" \( (>) \), "less than or equal to" \( (\le ) \) and "greater than or equal to" \( ( \ge ) \). For example, to compare the numbers 3 and 5, we can write \( 3 < 5 \) to indicate that 3 is less than 5.**Number line:**The number line is a visual representation of numbers on a line, with smaller numbers on the left and larger numbers on the right. To compare numbers on a number line, we can simply look at their positions relative to each other. For example, if we want to compare 3 and 5, we can see that 5 is to the right of 3 on the number line, so 5 is greater than 3.**Absolute value:**The absolute value of a number is the distance of that number from zero on the number line. To compare two numbers with the same sign, we can compare their absolute values. For example, to compare -3 and -5, we can compare the absolute values of these numbers, which are 3 and 5, respectively. Since 5 is greater than 3, we can say that The absolute value of -5 is greater than The absolute value of -3.**Algebraic manipulation:**We can use algebraic manipulation to compare expressions by simplifying them and comparing the resulting expressions. For example, to compare the expressions \(2x+3\) and \(3x-1\) , we can simplify them by combining like terms and get \( 2x+3 < 3x-1 \) . Then, we can isolate the variable on one side of the inequality and get \( x> 4 \) .**Common denominator:**When comparing fractions, we can find a common denominator and then compare the numerators. For example, to compare \(\frac{1}{4}\) and \(\frac{2}{5}\) , we can find a common denominator of 20 and get \(\frac{5}{20}\) and \(\frac{8}{20}\) . Then, we can see that \(\frac{8}{20}\) is greater than \(\frac{5}{20}\) , so \(\frac{2}{5}\) is greater than \(\frac{1}{4}\) .

These are just a few of the methods used to compare numbers and expressions in mathematics. The choice of method depends on the specific problem and the tools available.