Problem 52 - Elementary Algebra

Question

For the following problems, introduce a variable (any letter will do) and use appropriate algebraic symbols to write the given statement. A number divided by six is greater than or equal to forty-four.

Step-by-Step Solution

Answer

Question: Translate the statement "A number divided by six is greater than or equal to forty-four" into an algebraic inequality. Answer: x/6 ≥ 44

Step 1: Represent the number with a variable

Let x represent the unknown number.

Step 2: Write the expression for "divided by six"

Divide the unknown number x by six. This can be represented by the fraction x/6.

Step 3: Write the inequality for "greater than or equal to forty-four"

The expression "greater than or equal to" can be represented by the symbol ≥. So, the inequality becomes x/6 ≥ 44. Thus, the algebraic inequality representing the given statement is: x/6 ≥ 44.

Key Concepts

Algebraic Expressions

In the realm of math, algebraic expressions are fundamental blocks that allow us to capture mathematical ideas using numbers, variables, and operators. Think of an algebraic expression as a phrase that can include ordinary numbers, variables (like placeholders for numbers we don't know yet), and operations such as addition, subtraction, multiplication, and division.

For example, if I tell you that I have 'a number divided by six,' in algebra, this can be expressed as a fraction where the number is above the line (numerator), and 'six' is below the line (denominator). If we don't know the actual number, we can assign a variable to represent it. Typically, we use letters like 'x', 'y', or 'z' to stand in for these unknown numbers. Therefore, 'a number divided by six' becomes \( \frac{x}{6} \) in algebraic notation, where 'x' is our unknown number.

Inequality Symbols

Inequalities are like the siblings of equal signs; they tell us how numbers or expressions compare to each other rather than matching up equally. The common inequality symbols are: < (less than), ≤ (less than or equal to), > (greater than), and ≥ (greater than or equal to).

When we use these symbols, we're creating a statement about the relative size or value of two quantities. For instance, the inequality 'greater than or equal to,' written as ≥, is saying that one quantity is either more than or precisely equal to another. It's like saying, 'I have at least this many,' in everyday language, which includes the possibility of having exactly 'this many' or more. So, when an exercise asks for a representation of a number being 'greater than or equal to forty-four,' it is concisely written using the symbol ≥ as \( x ≥ 44 \) with no confusion about the meaning.

Variable Representation

Variables are the alphabet soup of mathematics; they're symbols that stand in for unknown numbers. When faced with a problem that talks about 'a number,' we use a variable to represent this unknown quantity because it gives us the flexibility to solve problems without needing specific values.

In the provided exercise, we chose the variable 'x' to stand for 'a number.' This choice is arbitrary— we could have chosen any letter of the alphabet. The key is consistency: once you choose a variable, stick with it throughout the problem. Variable representation is like naming a character in a story; it's how we refer to them in our mathematical narrative. So the phrase 'a number divided by six is greater than or equal to forty-four' transitions elegantly into the algebraic inequality \( \frac{x}{6} ≥ 44 \) when following these conventions.