Problem 107 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. The difference of 4 and -8

Step-by-Step Solution

Answer

8

Step 1: Identify the phrase components

The phrase involves 'twice the product of -1 and 6' and 'subtracted from -4'.

Step 2: Calculate the product

Find the product of -1 and 6, which is \(-1 \times 6 = -6\).

Step 3: Multiply the product

The phrase asks for 'twice' the product, which means multiplying the result by 2: \(2 \times -6 = -12\).

Step 4: Subtract the result from -4

Subtract the value obtained from -4: \(-4 - (-12) = -4 + 12 = 8\).

Key Concepts

Simplification

Simplification is the process of reducing a mathematical expression to its simplest form. It allows you to express complex calculations in a more straightforward way. Simplifying expressions involves various operations such as addition, subtraction, multiplication, and division.

For instance, in the original problem, the product of -1 and 6 was calculated first, and then the result was multiplied by 2. Simplifying helps us break down these steps:

First, calculate the product: \(-1 \times 6 = -6\).
Next, double the product: \(2 \times -6 = -12\).

• This makes the expression easier to handle.
• You get a clearer understanding of the problem.

Finally, combining this with subtraction helps simplify the whole exercise, making calculations and understanding much simpler.

Algebraic Expressions

An algebraic expression includes numbers, variables, and arithmetic operations. These expressions can be simplified to make problem-solving easier by combining like terms and using properties of operations.

In the provided exercise, we did not encounter variables but worked entirely with numerical values. However, the principles are similar. The exercise demonstrated how to combine operations in a clear sequence to solve the problem. Breaking it down step by step allows you to handle every part of the equation:

For example:

• Identify ‘twice the product’ from the phrase.
• Calculate the product of numbers involved.
• Perform the multiplication to find twice the product.
• Subtract this result from the given number.

This practice ultimately forms the foundation for dealing with more complex algebraic expressions involving variables.

Step-by-step problem solving

Step-by-step problem solving is a methodical approach to tackling mathematical problems. It involves breaking down a problem into smaller, more manageable steps. This strategy helps you understand and solve the problem efficiently.

In the exercise given:
1. We first identified the key components of the phrase.
2. Calculated the product as a separate step.
3. Multiplied the result to find ‘twice the product’.
4. Subtracted this result from the specified number.

This approach makes solving complex problems less daunting. A step-by-step method ensures you follow a logical sequence:

• Identify key terms or numbers.
• Perform operations in the correct order.
• Check each step to avoid mistakes.
• Gradually build up to the final solution.

Through this method, you can apply the same logical flow to both simple and complex mathematical problems, ensuring clarity and correctness in your work.