Explain to me why eight minus -10 equals 18 when -9+ -15 equals 34

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Let's start by understanding integers and how they work. Integers include all positive numbers, negative numbers, and zero. On a number line, positive numbers are to the right of zero, while negative numbers are to the left. When we perform operations with integers, we follow specific rules. Adding moves us to the right on the number line, while subtracting moves us to the left. Understanding these basics will help us solve more complex problems involving positive and negative numbers. Now let's understand why eight minus negative ten equals eighteen. The key is the double negative rule. When we subtract a negative number, it's the same as adding a positive number. So eight minus negative ten becomes eight plus ten, which equals eighteen. Let's see this on the number line. We start at eight, and instead of subtracting, we add ten because subtracting a negative becomes adding a positive. Moving ten steps to the right from eight brings us to eighteen. This is why the answer is eighteen, not negative two as some might think. Now let's look at adding negative numbers, which is different from subtracting them. When we add two negative numbers, we move further left on the number line, making the result more negative. Let's solve negative nine plus negative fifteen. We start at negative nine, then add negative fifteen, which means moving fifteen steps to the left. This brings us to negative twenty-four, not positive thirty-four as incorrectly stated in the original question. Adding negative numbers always results in a more negative number, never a positive one. Let's correct the misconception by comparing both problems side by side. The first problem, eight minus negative ten, correctly equals eighteen because subtracting a negative number is the same as adding a positive number. However, the second problem, negative nine plus negative fifteen, equals negative twenty-four, not thirty-four as incorrectly stated. This is a common mistake students make when they confuse the rules for different operations. Remember: subtracting a negative equals adding a positive, while adding negatives results in a more negative number. These are fundamentally different operations with different rules. Let's practice with more examples to solidify our understanding. First, five minus negative three equals eight, because subtracting a negative becomes adding a positive. Second, negative seven plus negative two equals negative nine, as adding negatives makes the result more negative. Third, twelve minus negative eight equals twenty, again applying the double negative rule. Fourth, negative four plus negative six equals negative ten. Each problem can be verified using the number line, where we visualize the direction and distance of movement. Remember the key rules: subtracting a negative means adding a positive, while adding negatives results in a more negative number.