Problem 112 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. The sum of 12 and \(-7,\) decreased by 14

Step-by-Step Solution

Answer

-30

Step 1: Identify key components

The phrase includes 'product of -3' and 'difference of 3 and -7'. Identify these key elements before forming the expression.

Step 2: Write the difference part

The difference between 3 and -7 can be written as: \[ 3 - (-7) \]

Step 3: Simplify the difference

Simplify the difference by removing the parentheses: \[ 3 - (-7) = 3 + 7 = 10 \]

Step 4: Formulate the product

The phrase asks for the product of -3 and the difference. The expression now becomes: \[ -3 \times 10 \]

Step 5: Calculate the product

Multiply -3 and 10 to get: \[ -3 \times 10 = -30 \]

Key Concepts

Product

In mathematics, the term 'product' refers to the result of multiplying two or more numbers together. It is denoted using the multiplication symbol (\(\times\)). In our exercise: 'The product of -3 and the difference of 3 and -7', the key part is the phrase 'the product of -3'. This indicates we will multiply -3 by another value. Understanding the concept of a product is crucial because it allows you to combine numbers to find their combined value through multiplication.

Difference

The 'difference' in mathematics refers to the result of subtracting one number from another. In this exercise, we need to find the difference between 3 and -7. The phrase 'the difference of 3 and -7' can be translated into a numerical expression: \(3 - (-7)\). This expression might look complex at first, but remember that subtracting a negative number is the same as adding its positive counterpart. Thus, \(3 - (-7) = 3 + 7\). The result is 10. Understanding the concept of difference is important because it helps you determine how much one number deviates from another.

Simplification

Simplification in mathematics means making an expression easier to work with. This often involves performing operations like addition or multiplication to find a single numerical result. In our exercise, after finding the difference as 10, the next step is to simplify the expression \(-3 \times 10\). To do this, multiply -3 by 10 to get -30. Following these steps of simplification helps you break down complex expressions into manageable parts and find the final answer efficiently.