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Understanding how subtracting negative numbers works is crucial to solving many mathematical problems. When you subtract a number, you typically reduce the quantity of something. However, when you're subtracting a negative number, it's akin to dealing with two minuses which result in a positive. For instance, when we see an expression like \( -4 - (-19) \) it may look confusing at first.

Imagine you owe someone 4 dollars, represented by \( -4 \), but then you find 19 dollars, shown as \( -(-19) \). Instead of owing more, you actually have more money than before. This is why subtracting a negative number effectively becomes adding the opposite (positive) of that number. Thus, the expression becomes \( -4 + 19 \) and simplifies to \( 15 \).

In summary, always remember that two negatives make a positive when dealing with subtraction. This simple switch from subtraction to addition can make evaluating such expressions much more straightforward.

Evaluating expressions involves performing the operations included in the expression in the appropriate order. When faced with an expression such as \( -4 - (-19) \), it's essential first to convert it into a simpler form by applying the rule for subtracting negative numbers, as discussed before.

An important step is rewriting the expression in an equivalent form that is easier to compute, like changing subtraction of a negative to addition of a positive. Once you have \( -4 + 19 \), you simply perform the addition. The process of evaluating becomes much easier with this clearer form. Breaking down the expression step-by-step and performing the arithmetic leads to the solution, which in this case, is \( 15 \).

Always approach expressions methodically, looking out for negative signs and the order of operations to correctly simplify and evaluate them. This technique is not just specific to integers but applies to algebraic expressions as well.

The addition of integers—both positive and negative—is a fundamental concept in mathematics. When adding integers, there are a few key points to remember.

Firstly, when the signs of the numbers are the same, you add their absolute values, keeping the sign:. For instance, \( -3 + -5 = -8 \). Conversely, when the signs are different, you find the difference between the absolute values and take the sign of the larger absolute value: for example, \( -4 + 19 \).

Adding Different Signs

With \( -4 + 19 \), you compare the absolute values (4 and 19). Since 19 is larger, your answer will be positive. Subtract the smaller absolute value from the larger (\[ 19 - 4 = 15 \]) and retain the sign of the larger value, giving you \(+15\).

By understanding the rules for the addition of integers, you can handle mathematical problems that involve combining different kinds of numbers—positive, negative, and zero. Practicing these rules will strengthen your ability to quickly and accurately perform calculations.