Problem 66 - Elementary Algebra

Question

Write an equivalent inequality. All real numbers less than or equal to zero.

Step-by-Step Solution

Answer

The equivalent inequality is \( x \leq 0 \).

Step 1: Understanding the Problem

The exercise asks for an inequality that represents "all real numbers less than or equal to zero." This means we need to express a condition using mathematical symbols to capture this relationship.

Step 2: Identifying the Inequality Symbol

The phrase "less than or equal to" translates to the inequality symbol \( \leq \). This symbol indicates that numbers are included if they are less than or equal to a certain value.

Step 3: Setting Up the Variable

We can use any variable to represent the unknown real number. Commonly, we use \( x \) to represent a number in algebra. So let \( x \) be the real number of interest.

Step 4: Constructing the Inequality

Given that the numbers are less than or equal to zero, we combine the variable and the symbol to form the inequality: \( x \leq 0 \). This inequality captures the entire set of numbers that are either zero or negative.

Key Concepts

Inequality Symbols

Inequality symbols are essential tools in mathematical expressions to describe relationships between numbers or expressions. Unlike an equation, which shows equality, an inequality reveals that one side is greater than, less than, or equal to the other. Here are the most common inequality symbols:

• The symbol \(<\) means "less than," indicating the number on the left is smaller than the number on the right.
• The symbol \(>\) means "greater than," expressing that the number on the left is larger than the number on the right.
• The symbol \(\leq\) represents "less than or equal to," which allows for values equal to or smaller than the number on the right.
• The symbol \(\geq\) stands for "greater than or equal to," permitting values equal to or larger than the number on the right.

Inequality symbols are critical in writing mathematical expressions that depict limitations or ranges on numbers.
They provide a clear way to signify constraints that cannot be captured simply by equations.

Real Numbers

Real numbers are a fundamental concept in mathematics. They include all the numbers that can be found on the number line.

• This set includes positive numbers, negative numbers, and zero.
• There are two main types of real numbers: rational and irrational numbers.

Rational numbers are those that can be expressed as a fraction of two integers. This group includes all integers, fractions, and repeating or terminating decimals.
On the other hand, irrational numbers cannot be precisely expressed as fractions, and their decimal expansions are non-repeating and non-terminating. Examples include \(\pi\) and \(\sqrt{2}\). Real numbers are significant because they encompass almost all numbers used in everyday calculations, measurements, and various scientific computations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations. They are used to represent quantities and relationships in a generalized form.

• Variables are symbols, typically letters like \(x\), \(y\), or \(z\), used to stand in for unknown or changeable values.
• Constants are fixed numerical values present within the expression.
• Operators such as \(+\), \(-\), \(\times\), and \(\div\) are used to link variables and constants in an expression.

For example, in the expression \(2x + 3\), \(2\) is a coefficient, \(x\) is a variable, and \(3\) is a constant. Algebraic expressions are the foundation for forming equations and inequalities.
They enable us to manipulate, solve, and find solutions for various mathematical problems by offering a way to generalize specific numbers or situations.