Problem 59 - Algebra 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the operation

The word 'quotient' indicates a division operation. So, the sentence is referring to \(x\) divided by 16.

Step 2: Identify the inequality

The words 'greater than or equal to' indicate an inequality, represented by the symbol \(\geq\). This tells us that the division from Step 1 should be greater than or equal to a certain value.

Step 3: Identify the numbers

The numbers '16' and '32' have been given. '16' is the divisor from the division operation in Step 1, and '32' is the value that the quotient is compared to.

Step 4: Write the inequality

Putting these elements together, the mathematical representation of the given sentence is \(\frac{x}{16} \geq 32\).

Key Concepts

Equations

Equations are mathematical statements showing two expressions are equal. They typically have a left and a right side separated by an equal sign (=). Equations often involve numbers, variables, and arithmetic operations like addition, subtraction, multiplication, or division. Understanding equations helps to find unknown values, known as variables.

• For example, in the equation \(2x + 3 = 7\), the goal is to determine the value of \(x\) that makes the equation true.
• Solving equations involves performing algebraic operations on both sides to isolate the variable.

Equations are foundational in solving real-world problems where relationships between different quantities need to be expressed and understood clearly.

Verbal Sentences

Verbal sentences describe mathematical relationships using words instead of symbols. They form the basis of math word problems and are crucial in translating everyday language into algebraic expressions. For instance, given the problem: "The product of \(x\) and 5 is 20", the verbal sentence needs to be converted into a formula.

• Keywords like 'product', 'sum', 'difference', 'quotient', and 'is' are signals for specific operations.
• 'Product' suggests multiplication, 'sum' means addition, 'quotient' implies division, and 'difference' is subtraction.

By recognizing these keywords, one can translate verbal expressions accurately into mathematical formats for further analysis and solution.

Mathematical Representation

Mathematical representation converts verbal sentences into symbols, equations, and expressions. This translation is vital as it allows problems to be solved using algebraic methods. In the exercise, converting "The quotient of \(x\) and 16 is greater than or equal to 32" into a mathematical statement is crucial for understanding and solving the problem.

• The term 'quotient' indicates division, leading to the expression \(\frac{x}{16}\).
• 'Greater than or equal to' is represented as \(\geq\).
• Finally, the number 32 concludes the inequality \(\frac{x}{16} \geq 32\).

Such conversions streamline the process of problem-solving by providing a clear and precise language that is universally understood in mathematics.