Question

Use inequality notation to describe the subset of real numbers.\(x\) is positive.

Short answer

The subset of real numbers where \(x\) is positive can be described by the inequality \(x > 0\).

Step-by-step solution

Identify the Condition

First, understand the condition in the problem, which is that \(x\) is supposed to be a positive real number.

Translate into Mathematical Inequality

Translate the condition into mathematical terms. In mathematical terms, a positive number is a number that is greater than zero. Therefore, \(x\) being positive can be represented as \(x > 0\).

Interpret in Words

Interpret the inequality in words. So, the inequality \(x > 0\) means that \(x\) is any real number greater than zero. This is our answer.

Key concepts

Real Numbers

Understanding what real numbers are is essential when dealing with inequalities and their notations. In mathematics, the set of real numbers includes all the numbers that can be found on the number line. This diverse group consists of rational numbers (like fractions and integers) and irrational numbers (like \(\sqrt{2}\) or \(\pi\)). Real numbers are significant because they're used to measure continuous quantities and can be as large or small, negative or positive, as imaginable.

For the task at hand, visualizing real numbers on a number line can be very helpful. When we say that \(x\) is a 'positive' real number, imagine all the points to the right of zero on this line. These points represent numbers greater than zero. It's this visual cue that aids in comprehending and solving equations and inequalities involving real numbers.

Mathematical Inequality

Now, let's explore the concept of a mathematical inequality. Think of an inequality as a balance scale where each side doesn't necessarily have to be equal. They are a way to compare two expressions or values, stating that one is larger or smaller than the other. Inequalities are an integral part of mathematics because they help us describe ranges of possible values rather than a single definite value.

Common inequality symbols include \(>\) for 'greater than', \(<\) for 'less than', \(\geq\) for 'greater than or equal to', and \(\leq\) for 'less than or equal to'. When we say \(x > 0\), we are using inequality notation to articulate that \(x\) is any value greater than zero, but not equal to zero. Notably, inequality notation conveys a broad, continuous set of numbers that satisfy the condition, unlike an equation that often specifies exact numbers.

Solving Inequalities

Solving inequalities is a critical skill in mathematics. The process starts with understanding the conditions presented. Then, one translates those conditions into a mathematical representation using inequalities, as shown in the step-by-step solution for the exercise involving \(x\) being positive. The approaches for solving inequalities closely mirror those for solving equations—such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number.

However, a unique rule comes into play: if you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality. This means that if you had \(x > 0\) and you multiplied both sides by -1, the inequality would become \(x < 0\). Solving inequalities often results in a range or set of numbers that fulfill the condition, and understanding how to express these solutions, usually by graphing on a number line or using interval notation, is crucial for thorough mathematical comprehension.