Problem 69 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation or an inequality. Let x represent the number. 9 times a number is less than 6

Step-by-Step Solution

Answer

The inequality for the given sentence is \(9x < 6\).

Step 1: Identify and represent the unknown number

In the given sentence, 'a number' is the unknown piece of information which can be represented using a variable 'x'. So, any time 'a number' is mentioned, it can be swapped out with 'x'.

Step 2: Construct the expression for '9 times a number'

The phrase '9 times a number' can be translated to '9 times x' in mathematical terms, which becomes \(9x\). This represents multiplication of the unknown number by 9.

Step 3: Translate 'is less than 6' into mathematical notation

The phrase 'is less than 6' can be expressed as '< 6' in mathematical notation. It signifies that the result of '9 times the number' is less than 6.

Step 4: Formulate the complete inequality

By placing both of the translated phrases (from steps 2 and 3) together, the complete inequality becomes \(9x < 6\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental component of mathematics that involves combinations of numbers, variables, and operators such as addition, subtraction, multiplication, and division. In the exercise, the phrase "9 times a number" is an example of an algebraic expression. Here, "a number" is unknown, which is why we represent it using a variable, often denoted by letters like 'x', 'y', or 'z'. Variables are placeholders for values that may change or values we want to find.
To craft an algebraic expression, we identify operations mentioned in the verbal statement. In this case, 'times' indicates multiplication, resulting in the expression "9x," meaning 9 multiplied by the variable 'x'. This simplicity allows algebraic expressions to express complex ideas succinctly.

Inequality Notation

Inequality notation is used to compare two values or expressions, showing that one is greater, lesser, or potentially equal to the other. In the exercise, we see the term "is less than," which is represented by the symbol "\(<\)."
There are several inequality symbols, each signifying a different relationship between the quantities involved:

• "\(<\)" means less than.
• "\(>\)" means greater than.
• "\(\leq\)" means less than or equal to.
• "\(\geq\)" means greater than or equal to.

Using these symbols helps in constructing precise mathematical statements quickly. Each plays a crucial role in forming inequalities, as they define the conditions under which the inequality holds true. In the sentence provided, "9 times a number is less than 6" is translated to the inequality \(9x < 6\) using the appropriate inequality notation.

Translating Sentences into Equations

Translating sentences into equations involves taking verbal descriptions of mathematical relationships and converting them into concise mathematical statements. This skill is important not only in algebra but throughout mathematics, as it allows us to model real-world situations accurately.
Let's see how to effectively translate a sentence like the one in our exercise: "9 times a number is less than 6." We start by identifying key components in the sentence: '9 times a number' becomes the algebraic expression \(9x\) using multiplication. Next, 'is less than 6' translates to the inequality \(< 6\), reflecting the relationship described.
By combining these translations, the full statement is \(9x < 6\). Mastering this type of translation requires practice and attention to detail, as each word or phrase in a sentence can change how the equation or inequality is formed.