Problem 58 - Algebra 1: Concepts and Skills

Question

Determine whether the statement is true or false. Use the subtraction rule or a number line to support your answer. If you subtract a negative number from a positive number, the result is always a positive number.

Step-by-Step Solution

Answer

The statement is true. Subtracting a negative number from a positive number always gives a positive result.

Step 1: Understanding the Statement

The statement 'If you subtract a negative number from a positive number, the result is always a positive number' can be mathematically represented as \(a - (-b) = c\), where a and b are positive numbers. The key here to understand is that subtraction of a negative number can be represented as addition of a positive number. Hence, the above statement can be represented as \(a + b = c\), where c is supposed to be a positive number as per the statement.

Step 2: Testing the Statement

In this step, let's verify the statement by substituting a and b with positive numbers. Let's consider a = 5 and b = 3. On substituting these values in equation \(a + b = c\), we get c = 5 + 3 = 8, which is a positive number.

Step 3: Fact Verification

Through the illustration in step 2, we can verify that by subtracting a negative number from a positive number, we always get a positive number. As mathematically, subtracting a negative number is equivalent to adding a positive number. Thus, the statement given in the problem is true.

Key Concepts

Positive and Negative Numbers

Understanding positive and negative numbers is fundamental in mathematics, especially when dealing with operations like subtraction and addition. A positive number is represented by a numeral without any sign or a plus sign before it and is located to the right of zero on a number line. In contrast, a negative number has a minus sign before it and is found to the left of zero on the number line.

The crucial concept to remember with negative numbers is that they represent a value less than zero, often used to denote a deficiency or loss. When subtracting negative numbers, it's as if you're removing a debt, thus adding value to the overall quantity, moving you towards the positive side of the number line. This is why subtracting a negative number is equivalent to adding the absolute value of that same number. For instance, if you have \( -5 \) and you subtract \( -3 \) from it, effectively you are adding 3, resulting in \( -2 \) instead of a further decreased value. This strongly counters our usual perception of 'subtract' as always 'decreasing' a quantity.

Subtraction Rule

The subtraction rule when dealing with negative numbers might seem counterintuitive at first. Normally, we perceive subtraction as taking something away, which would ostensibly lessen the quantity. However, the rule changes its common interpretation when it involves negative numbers.

The key rule to remember for subtraction is: subtracting a negative number is the same as adding its positive counterpart. Mathematically, it is expressed as \( a - (-b) = a + b \). This means that if you start with a certain amount, \( a \), and you 'subtract' a negative amount, \( -b \), you are actually adding \( b \) to \( a \). This can be visually demonstrated through the use of a number line or algebraically through the manipulation of signs. It's crucial to get comfortable with this rule, as it applies to all situations involving the subtraction of negative numbers, not just specific cases, ensuring a positive result when starting with a positive value.

Number Line

The number line is an excellent tool for visualizing and understanding the concept of subtraction, especially when it involves both positive and negative numbers. It's a straight line with marks at equal intervals, representing numbers in an ascending order from left to right. At the center of this line is zero, the point of reference separating negative numbers on the left from positive numbers on the right.

Consider the statement from the exercise: subtracting a negative number from a positive number. When using a number line, if you start at a positive number and want to subtract a negative number, you actually move to the right because you are adding the absolute value of the negative number. For example, starting at 5 on the number line and subtracting -3 would involve moving 3 units to the right, resulting in landing on 8. This graphical representation helps solidify the understanding that subtracting a negative number results in an increase in value—a concept that students often find tricky when first encountered.