Problem 65 - Elementary Algebra

Question

Write an equivalent inequality. All real numbers less than \(27 .\)

Step-by-Step Solution

Answer

The equivalent inequality is \( x < 27 \).

Step 1: Understand the Inequality

We need to write an inequality expression that captures all real numbers less than 27. This means we're looking for an inequality that includes every number below 27.

Step 2: Formulate the Inequality

To represent all numbers less than 27, we can use the inequality symbol "<". Therefore, the inequality for this situation is \( x < 27 \).

Step 3: Verify the Inequality

Check whether the inequality correctly includes numbers that are less than 27. For instance, if \( x = 25 \), then \( 25 < 27 \), confirming that our inequality accurately represents the condition.

Key Concepts

Understanding Real Numbers

Real numbers are a foundational concept in mathematics, encompassing all the numbers that you can find on the number line. This includes:

• Whole numbers, like 5 or 10
• Positive and negative numbers, such as -3 and 7
• Fractions, like \( \frac{1}{2} \) or \( \frac{3}{4} \)
• Decimals, such as 2.5 or -0. 75
• Irrational numbers, like \( \pi \) and the square root of 2

Real numbers are infinite, meaning there's no end to the possibilities between any two numbers on the line. For example, between 0 and 1, there are countless numbers like 0.1, 0.01, and so on.
Real numbers are essential in describing quantities and values in real-world contexts, and they are crucial in mathematical calculations.

Crafting Inequality Expressions

Inequality expressions are used in mathematics to represent relationships between two values where they are not equal. An inequality expression involves variables and numbers and indicates how they compare, such as greater than or less than.
Creating inequality expressions involves using this format:

• A variable (like \( x \))
• An inequality symbol (like \(<\) for less than)
• A number (such as 27 in our example)

Inequality expressions can represent real-world situations like temperatures falling below freezing or limiting speeds on a highway.

Inequality Symbols: Key to Comparison

Inequality symbols are shorthand notations used to describe the relationship between two values or expressions. These symbols are:

• "<" for "less than"
• ">",