Problem 52 - Algebra for College Students

Question

Simplify each numerical expression. $$6-9+11-8-7+14$$

Step-by-Step Solution

Answer

The simplified expression equals 7.

Step 1: Understand the Expression

We have the numerical expression: \(6 - 9 + 11 - 8 - 7 + 14\). We need to simplify this expression by performing all additions and subtractions in sequence from left to right.

Step 2: Subtract 9 from 6

First, solve the first operation, \(6 - 9\), which equals \(-3\).

Step 3: Add 11

Continue by adding 11 to the result from Step 2. \(-3 + 11 = 8\).

Step 4: Subtract 8

Now subtract 8 from 8. \(8 - 8 = 0\).

Step 5: Subtract 7

Continuing with the expression, subtract 7. \(0 - 7 = -7\).

Step 6: Add 14

Finally, add 14 to the result from Step 5. \(-7 + 14 = 7\).

Key Concepts

Integer Operations

Integer operations are a fundamental part of mathematics, involving the basic arithmetic processes - addition, subtraction, multiplication, and division.
When dealing with integers, it's important to remember that they include positive numbers, negative numbers, and zero. Understanding how operations work with these numbers is essential for simplifying numerical expressions.

• Adding Integers: To add two integers, pay attention to the signs. If the integers have the same sign, add their absolute values and keep the common sign.


• Subtracting Integers: When subtracting, think of subtracting a number as adding its opposite. For example, subtracting -3 is the same as adding +3.


• Negative numbers: Remember, for operations involving negative numbers:
  • A positive number added to a negative number results in a lower sum.
  
  
  • A negative number subtracted from another negative gives a positive result if the subtraction happens from a larger number.

Understanding these basics helps when simplifying expressions involving several integer operations.

Order of Operations

The order of operations is a set of rules that indicates the sequence in which operations should be performed. When simplifying expressions, it's essential to follow these rules to achieve accurate results.
For most basic arithmetic without parentheses or complex expressions, this involves simplification from left to right for additions and subtractions.

• Left-to-right Rule: In our practice expression, notice how the operations are carried out:
  • Subtract first, then add, following the sequence of the numbers regardless of the operation type.
  
  


• Examples: Consider the expression:
  • 6 - 9 was performed first.
  
  
  • Then, addition, subtraction, and so on, in the order they appear from left to right.

By adhering to these principles, you prevent errors and ensure mathematical expressions are solved correctly.

Addition and Subtraction

Addition and subtraction are basic arithmetic operations that are often performed together in a sequence when simplifying expressions.
Since subtraction can be seen as adding a negative number, sometimes these operations are combined within the same process.

• Addition: Combines two numbers to form a larger value.
  • For example, in the expression step "-3 + 11", adding resulted in 8.
  
  


• Subtraction: Involves taking one number away from another, which can decrease the value or turn it negative.
  • In the step "8 - 8", we could see subtraction leading the result to zero.
  
  
  • Later, "0 - 7" reduced the number to -7.

Recognizing how addition and subtraction interact allows for smoother progression through steps when simplifying expressions.