91Ó°ÊÓ

Inequalities are similar to equations, but instead of an equal sign, they use symbols such as <, >, ≤, or ≥. The key objective when solving inequalities, like equations, is to find the value or range of values for the variable that makes the inequality true. Every time you perform an operation to one side of the inequality, you must do the same to the other side to maintain equality. This includes addition, subtraction, multiplication, or division.

For example, given the inequality \(-\frac{3}{5} x < -\frac{6}{7}\), we want to isolate \(x\) by performing operations that will leave \(x\) alone on one side. When performing operations on inequalities, a special rule applies when using multiplication or division on both sides - particularly involving negative numbers. We'll discuss that in the next section.

Multiplying or dividing both sides of an inequality by a negative number requires a special step: reversing the inequality sign. This step is crucial because the order of numbers changes when you multiply or divide by a negative.

Consider the inequality \(-\frac{3}{5} x < -\frac{6}{7}\). To isolate \(x\), we multiply both sides by \(-\frac{5}{3}\) (the reciprocal of \(-\frac{3}{5}\)). This operation results in flipping the inequality symbol from < to >:

• Initial: \(-\frac{3}{5} x < -\frac{6}{7}\)
• After multiplying by \(-\frac{5}{3}\): \(x > \frac{10}{7}\)

This reversal ensures that the logic of the inequality is preserved when rounding through these calculations. Remember this key step when you encounter negatives in inequalities.

Real numbers include all the numbers that are on the number line. This set is comprehensive, incorporating integers, zero, fractions, decimals, and irrational numbers. Understanding the nature of real numbers helps us appreciate the range of possible solutions an inequality might have.

In our exercise, the solution to the inequality \(x > \frac{10}{7}\) implies \(x\) is any real number greater than \(\frac{10}{7}\). The answer is expressed in set-builder notation, which is a formal way to describe sets defined by a property that its members must satisfy. It gives us:

• \(\{x \mid x \in \mathbb{R}, x > \frac{10}{7}\}\)

This notation encompasses all possible values of \(x\) that exceed \(\frac{10}{7}\), ensuring every potential solution is accounted for in the realm of real numbers.