Problem 72 - Algebra for College Students

Question

Simplify each numerical expression. $$(7-12)(-3-2)$$

Step-by-Step Solution

Answer

The expression simplifies to 25.

Step 1: Simplify the First Expression in Parentheses

The first part of the expression is \(7 - 12\). Subtract 12 from 7 to simplify: \(7 - 12 = -5\).

Step 2: Simplify the Second Expression in Parentheses

The second part of the expression is \(-3 - 2\). Add -2 to -3 to simplify: \(-3 - 2 = -5\).

Step 3: Multiply the Simplified Values

Now that you have simplified both parts to \(-5\), multiply them together: \((-5) \times (-5) = 25\).

Key Concepts

Numerical Expressions

Numerical expressions are mathematical phrases that involve numbers and operations. They do not include any variables. Think of them like a sentence in math; instead of words, you have numbers and operators (like +, -, *, /). These expressions require simplification to reach a concise result.
In our example, the expression is \((7 - 12)(-3 - 2)\). This expression involves subtraction and multiplication. Each set of parentheses encapsulates a separate numerical expression that should be simplified before performing multiplication.
When dealing with numerical expressions, it is crucial to follow the order of operations (often remembered by PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to ensure accuracy in the simplification process.

Simplification Process

Simplification is the process of making an expression easier to work with by reducing it to a simpler form. For numerical expressions, this often involves performing arithmetic operations until you are left with a single value or a simple product.
The simplification process follows a series of steps:

• Identify and solve operations within parentheses.
• Follow the operations as per the mathematical hierarchy (order of operations).
• Perform basic arithmetic such as addition, subtraction, multiplication, or division.

In our exercise, we simplified the numbers inside the parentheses first:

• The expression \(7 - 12\) became \(-5\) because subtraction was performed.
• Similarly, \(-3 - 2\) was simplified to \(-5\) by recognizing it as an addition of negative numbers.

This simplification made it easy to multiply the results further.

Multiplication of Integers

Multiplication of integers involves combining two or more whole numbers (which can be negative or positive) into a single product. An important rule when multiplying integers is that the product of two negative numbers is a positive number.
In our given problem, the integers \(-5\) and \(-5\) are multiplied:

• Multiply the absolute values: \(5 \times 5 = 25\).
• Apply the sign rule: Since both numbers are negative, the product is positive.

This gives us the final answer of \(25\). Understanding these core principles helps in confidently solving similar problems and recognizing the significance of integer signs in operations. Integers follow specific rules that can simplify working with them and making calculations straightforward.