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Positive and negative numbers are the basic building blocks in mathematics. Positive numbers are greater than zero, while negative numbers are less than zero.

Imagine numbers lined up on a number line: positive numbers go to the right of zero, and negative numbers to the left. Zero is neither positive nor negative.

When dealing with integers, an important aspect is recognizing how two numbers interact. For example, a negative number means you owe something, while a positive number indicates you have something extra.

Operations involving these numbers, like adding or multiplying, depend on understanding how positives and negatives combine. This will aid in simplifying complex expressions without errors.

Multiplying integers may seem simple, but the rules change when dealing with negatives.

Here are some basic principles:

• Multiplying two positive numbers results in a positive number. For instance, \(3 \times 4 = 12\).
• Multiplying two negative numbers also results in a positive number. For example, \((-2) \times (-3) = 6\). This happens because the negatives cancel out, like two wrongs making a right.
• When multiplying a positive number by a negative number, the result is negative. For instance, \(5 \times (-7) = -35\), and vice versa \((-9) \times 6 = -54\).

Recognizing these rules helps in quickly determining the sign of a product, like in our exercise examples.

Dividing integers entails similar rules to multiplying. Understanding these can help prevent mistakes.

When dividing:

• If both integers share the same sign, the quotient is positive. For example, dividing two negatives, such as \((-8) \div (-2)= 4\).
• If the integers have different signs, the quotient is negative. For example, \(18 \div (-3) = -6\).

Division essentially reverses multiplication processes, where the rules about signs hold consistently, even with fractions or zero. Remember not to divide by zero, as it makes calculations undefined. In the given exercise, this clarity helps predict outcomes without solving every step.

Adding and subtracting integers can often be simplified by thinking about each operation as movements on a number line. Consider:

• To add a positive number, move right on the number line.
• To add a negative number, move left.
• Subtracting a positive number is the same as adding a negative, and subtracting a negative is the same as adding a positive. This is often summed up by saying "subtracting a negative is like addition".

For example, consider \(25 - \frac{152}{12}\). Here, you handle division first, acknowledging the relative values and number line shifts. Once visualized, these concepts incorporate naturally into solving exercises, minimizing calculator dependence.