91Ó°ÊÓ

Inequalities represent the relative size or order of two values. Unlike equations, which show equality, inequalities demonstrate that one quantity is larger or smaller than another. They are signified by symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). For example, if we have two numbers, A and B, the inequality A < B tells us that A is less than B. Similarly, A > B indicates that A is greater than B.

When dealing with inequalities and variables, just like in the provided exercise \( -\frac{4}{5} x< -\frac{8}{15} \), our objective is to find the set of values that make the inequality true. This concept is fundamental in mathematics, as it is used to describe a range of possible solutions rather than a single solution as with equations.

Solving inequalities involves finding all the values of the variable that makes the inequality true. The principles are similar to solving equations—performing equivalent operations on both sides to isolate the variable—except for a critical distinction concerning multiplying or dividing by negative numbers. When you multiply or divide both sides of an inequality by a negative number, you must reverse the symbol of the inequality.

For instance, in the step by step process you mentioned, the variable \(x\) was multiplied by \( -\frac{4}{5} \) and to isolate \(x\), division by the same negative number was performed. This action reversed the inequality from '<' to '>', converting the original expression \( -\frac{4}{5} x < -\frac{8}{15} \) into \(x > \frac{8}{15} / -\frac{4}{5}\). The same principle applies when multiplying by a negative number; the direction of the inequality must be reversed. This rule is essential to remember as it directly affects the range of solutions for the variable.

Fraction division can initially seem daunting, but with a grasp of the concept, it becomes straightforward. To divide fractions, we multiply the dividend (the fraction being divided) by the reciprocal of the divisor (the fraction by which we are dividing). The reciprocal of a fraction is obtained by exchanging the numerator and the denominator.

In the provided example, the divisor is \( -\frac{4}{5} \) whose reciprocal is \( -\frac{5}{4} \). The division \( \frac{8}{15} / -\frac{4}{5} \) is therefore equivalent to the multiplication \( \frac{8}{15} * -\frac{5}{4} \). When we multiply these fractions, we multiply the numerators to get the new numerator and multiply the denominators to get the new denominator for our fraction. Simplifying the result gives us the solution for the variable \(x\), which in this case is \( x > \frac{10}{3} \). Understanding how to properly divide fractions is not only valuable for solving inequalities but is a key skill in various areas of mathematics.