Problem 71 - Algebra for College Students

Question

Simplify each numerical expression. $$(6-11)(4-9)$$

Step-by-Step Solution

Answer

(6-11)(4-9) simplifies to 25.

Step 1: Evaluate the First Expression

Begin by evaluating the expression \(6 - 11\). Subtract 11 from 6 to get:\[6 - 11 = -5\]

Step 2: Evaluate the Second Expression

Next, evaluate the expression \(4 - 9\). Subtract 9 from 4 to obtain:\[4 - 9 = -5\]

Step 3: Multiply the Results

Now, multiply the results from Step 1 and Step 2. Multiply \(-5\) by \(-5\):\[(-5) \times (-5) = 25\]

Step 4: Conclusion

The simplified result of the expression \((6 - 11)(4 - 9)\) is 25.

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making complex mathematical statements easier to handle by breaking them down into more straightforward components. It is very much like tidying up your room; everything has to be sorted neatly. When we simplify, we aim to write it in the simplest form. In algebraic expressions, this typically involves reducing expressions to fewer terms or a single term.

Here is a brief guide:

• First, address any subtraction or addition within parenthesis by resolving it completely.
• Simplify inside out: handle what's inside the parenthesis before moving to other operations such as multiplication or division.
• Combine like terms by adding or subtracting them.
• Keep an eye on the signs; they are crucial, especially with negative numbers.

In our exercise, simplifying meant resolving the subtraction in each set of parentheses before multiplying the results. This approach helps to avoid potential errors and ensures the clarity of the work.

Numerical Expressions

Numerical expressions consist entirely of numbers and operation symbols, without any variables. They often require simplifying to reach a solution or to be expressed in a simpler form. Think of these expressions as tightly knit puzzles that need to be unwrapped and solved step-by-step.

To simplify numerical expressions, follow these guidelines:

• Identify and evaluate expressions or calculations within parentheses first as they are given precedence (order of operations).
• Use the basic operations: addition, subtraction, multiplication, and division to simplify.
• Always follow the correct order of operations, commonly remembered by the acronym "PEMDAS" (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).

The expression \( (6-11)(4-9) \) in our example is a numerical expression made simpler by following these rules step by step.

Negative Numbers

Negative numbers represent values less than zero and often appear in mathematical expressions. Understanding how they interact with each other and with other numbers is essential for simplifying expressions correctly. They can be tricky because they change the way we normally think about addition, subtraction, and multiplication.

Remember these basic rules when dealing with negative numbers:

• Subtracting a number is equivalent to adding its negative.
• Multiplying two negative numbers always results in a positive number.
• A positive number multiplied by a negative number results in a negative number.
• The opposite is true too: a negative number times a positive number is also negative.

In our exercise, we multiplied two negative results from within our expressions to get a positive product. Bearing these rules in mind helps us avoid confusion, especially in more complex assemblages of numbers.