Problem 15 - Elementary Algebra

Question

Translate the following into a mathematical statement. Negative eight is less than or equal to zero.

Step-by-Step Solution

Answer

\(-8 \leq 0\)

Step 1: Identify Key Phrases

The phrase 'Negative eight' refers to the number \(-8\), and 'is less than or equal to' is an inequality.

Step 2: Write the Expression

Combine the identified elements: the inequality symbol for 'is less than or equal to' is \(\leq\). Therefore, the expression becomes \(-8 \leq 0\).

Step 3: Verify the Interpretation

Check if \(-8\) being less than or equal to \(0\) makes logical sense. Yes, \(-8\) is indeed less than zero.

Key Concepts

Mathematical Statements

A mathematical statement is an expression that conveys a particular idea or relationship using numbers and symbols. It's like a sentence in regular language, but it uses math symbols to deliver its message.

• Statements with numbers relate numerical values to each other using operators or connectors.
• These statements can be an equation, like "3 + 5 = 8," or an inequality, like "5 < 10."

The goal is to clearly communicate a mathematical concept or fact. In the problem's example, "Negative eight is less than or equal to zero," we are using math to describe how the number -8 relates to zero. The translation into the mathematical form requires understanding the words and their mathematical symbols.

Inequality Symbols

Inequality symbols are essential in mathematics for comparing values. Unlike equal signs which show exact equality, inequality symbols demonstrate how values relate beyond exactness.

• Less than ( < ): Indicates a number is smaller than another (e.g., 2 < 5).
• Less than or equal to ( \(\leq\) ): Shows a number is either less than or equal to another (e.g., 3 \(\leq\) 3 or 3 \(\leq\) 5).
• Greater than ( > ): Used when a number is larger than another (e.g., 7 > 4).
• Greater than or equal to ( \(\geq\) ): Demonstrates a number is either greater than or equal to another (e.g., 5 \(\geq\) 5 or 6 \(\geq\) 4).

Inequalities help express mathematical statements where not all parts of the relation are equal, providing more depth beyond equations. In our exercise, we used the \(\leq\) symbol to indicate that -8 is less than or equal to 0.

Negative Numbers

Understanding negative numbers is crucial as they represent quantities less than zero and appear in various mathematical contexts.

• Negative numbers are typically used to denote things like debts or temperatures below zero.
• They are represented with a minus sign (-) in front, such as -1, -8, -15.

On the number line, negative numbers are located to the left of zero. The further left the number is, the smaller it is. For instance, -8 is smaller than both -1 and 0. Recognizing this helps when working with inequalities as it affects the direction and validity of the inequality. In the exercise example, knowing -8 is smaller helps justify the inequality statement \(-8 \leq 0\).