Problem 63 - Algebra 1

Question

EQUATIONS AND INEQUALITIES Match the verbal sentence with its mathematical representation. The quoticnt of \(x\) and 16 is greater than 32.

Step-by-Step Solution

Answer

The mathematical representation of the given sentence is \(\frac{x}{16} > 32\).

Step 1: Identify mathematical operations

Identify the mathematical operations in the sentence. The word 'quotient' suggests division, 'is greater than' refers to an inequality.

Step 2: Translate the sentence into a mathematical representation

Translate the words into mathematical representation. 'The quotient of \(x\) and 16' translates to \(\frac{x}{16}\), 'is greater than' translates to \(>\), and '32' stays as 32.

Step 3: Assemble mathematical representation

Now, put the translated segments together. This gives us the inequality \(\frac{x}{16} > 32\).

Key Concepts

Mathematical Operations

Mathematical operations are essential tools that help us solve problems and represent situations mathematically. In our exercise, we encounter two key operations: division and an inequality comparison.

The term 'quotient' refers to the result of dividing one quantity by another. When we see the word 'quotient,' it signals that division is the operation involved. In our case, the division is performed with an unknown value, "x," and the number 16:

• The phrase "the quotient of x and 16" translates to the mathematical expression \(\frac{x}{16}\).

The next operation is the inequality "is greater than," a statement indicating that one quantity exceeds another:

• 'Is greater than' is symbolized by the '>' sign in mathematics.
• This operation compares the result \(\frac{x}{16}\) with another number, which is 32 in this context.

Translating Verbal Sentences

Translating verbal sentences into mathematical expressions is a crucial skill in algebra. It involves understanding the language of math and deciphering words into symbols and operations that can be mathematically manipulated.

Let's break down the process using our example sentence: "The quotient of \(x\) and 16 is greater than 32."

• The word 'quotient' indicates division, guiding us to represent the part "the quotient of \(x\) and 16" as \(\frac{x}{16}\).
• 'Is greater than' is a phrase that tells us to use the inequality symbol ">," showing a comparison where the left side is larger.
• The number '32' is straightforward in translation—it's a constant, and we use it as is.

Combining these elements, we transform the entire verbal sentence into a concise mathematical statement: \(\frac{x}{16} > 32\). This methodical translation enables us to solve the inequalities or equations presented by the problem.

Mathematical Representation

The next step in solving algebraic inequalities involves assembling the mathematical representation. This requires organizing the translated segments into one clear statement that encompasses the entire relationship described in the verbal sentence.

After identifying and translating the separate parts of the sentence, we combine them to form a meaningful mathematical inequality. From our previous steps, we know that:

• "The quotient of \(x\) and 16" is \(\frac{x}{16}\).
• "Is greater than" directs us to the symbol ">."
• Finally, "32" remains as 32.

Putting these parts together results in the inequality \(\frac{x}{16} > 32\). This inequality expresses the original sentence in a mathematical form, making it ready for further analysis or solution.

By mastering mathematical representation, you can convert language into symbols that reflect the true meaning of a problem, facilitating easier manipulation and solution in algebra.