Problem 38 - Algebra A Combined Function

Question

Translate each phrase to an expression and simplify. Find the difference between -17 and -1 .

Step-by-Step Solution

Answer

The difference is -16.

Step 1: Identify the operation

The word 'difference' indicates that we need to use subtraction.

Step 2: Write the expression

Write the numbers you are finding the difference between in the order they appear in the phrase. The expression becomes: \(-17 - (-1)\)

Step 3: Simplify the expression

Recognize that subtracting a negative is equivalent to adding the positive. So, \(-17 - (-1)\) becomes \(-17 + 1\).

Step 4: Calculate the result

Perform the addition. Starting from -17 and moving 1 unit to the right on the number line, we reach -16. Thus, \(-17 + 1 = -16\).

Key Concepts

Subtraction

Subtraction is one of the basic arithmetic operations. It involves finding the difference between numbers. In simple terms, when you subtract, you take away a certain number of elements from a set. For example, if you have 5 apples and you subtract 2 apples, you are left with 3 apples.
In algebraic expressions, subtraction follows similar rules. Here are some key points about subtraction:

• Order Matters: Ensure you write numbers in the correct order since subtraction is not commutative. For example, 5 - 3 is not the same as 3 - 5.
• Identify Key Terms: Words indicating subtraction include 'minus', 'difference', 'less than'.
• Relation to Addition: Subtraction can be seen as the inverse of addition.

Understanding subtraction in both numerical and algebraic contexts is crucial for simplifying expressions accurately.

Negative Numbers

Negative numbers can often be tricky, especially when combined with subtraction. Negative numbers represent values less than zero and are crucial in various mathematical contexts.

• Behavior: When you subtract negative numbers, it alters the operation into an addition.
• Real World: Negative numbers often represent losses or decreases, such as temperatures below zero.
• Symbol: Negative numbers are denoted with a minus sign (-) before the number, like -5.

For instance, in our exercise, when we see \(-17 - (-1)\), the minus a negative turns into an addition, leading us to \(-17 + 1\).
It's essential to get comfortable with the concept of negative numbers to simplify expressions effectively.

Simplification

Simplification in algebra is a process where an expression is made easier to understand. It involves condensing terms and making computations straightforward.

• Identify Operations: Look for operations within the expression like addition, subtraction, multiplication, or division.
• Combine Like Terms: This means grouping similar terms together to reduce complexity.
• Recognize Patterns: Identifying patterns can help in quickly simplifying expressions.

In our problem \(-17 - (-1)\), we simplified it by acknowledging that subtracting a negative is the same as addition. Thus, it was simplified to \(-17 + 1\).
Mastering simplification can help in solving equations faster and more efficiently.

Number Line

A number line is a visual representation of numbers in a straight path. It's helpful in understanding addition, subtraction, and placement of numbers.

• Visual Aid: It provides a clear way to see the size and position of numbers.
• Direction Matters: Moving to the right signifies an increase, while moving left indicates a decrease.
• Uses: It helps in understanding operations like subtraction, particularly when negative numbers are involved.

In our example, finding the difference between \(-17\) and \(-1\) required understanding their positions on the number line.
We start at -17 and move 1 unit to the right to reach -16, visually representing the subtraction \(-17 + 1\).
Number lines are valuable tools in visually comprehending the impact of arithmetic operations.