1.4 Multiply and Divide Integers

Multiplying and dividing integers is one of the core skills you'll use throughout algebra. These operations follow specific sign rules, and once you've got those rules down, simplifying expressions and solving equations becomes much more straightforward.

Multiplying and Dividing Integers

Multiplication of positive and negative integers

The sign rules for multiplication are simple. There are really only two things to remember:

• Same signs → positive result
  • Positive × positive = positive: $2 \times 3 = 6$
  • Negative × negative = positive: $(-4) \times (-5) = 20$
• Different signs → negative result
  • Positive × negative = negative: $3 \times (-2) = -6$
  • Negative × positive = negative: $(-5) \times 4 = -20$

To find the product, multiply the absolute values of the two numbers, then apply the sign rule. In other words, $|a \times b| = |a| \times |b|$. For example, $|{-3} \times 4| = 3 \times 4 = 12$, and since the signs are different, the answer is $-12$.

One more thing: multiplying any integer by 1 leaves it unchanged. That's the multiplicative identity property, and it comes up more than you'd expect when simplifying expressions.

Division operations with integers

Division follows the exact same sign rules as multiplication:

• Same signs → positive result
  • $10 \div 5 = 2$
  • $(-12) \div (-3) = 4$
• Different signs → negative result
  • $15 \div (-3) = -5$
  • $(-20) \div 4 = -5$

There's one critical restriction: division by zero is undefined. You cannot divide any number by zero. $8 \div 0$ doesn't equal zero; it simply has no answer. If you see division by zero in a problem, the expression is undefined.

Properties of Integer Operations

These three properties show up constantly when you're simplifying and rearranging expressions:

• Commutative property: You can swap the order of factors without changing the product. $a \times b = b \times a$
• Associative property: You can regroup factors without changing the product. $(a \times b) \times c = a \times (b \times c)$
• Distributive property: Multiplication distributes over addition and subtraction. $a(b + c) = ab + ac$

Note that division is not commutative or associative. $10 \div 2$ is not the same as $2 \div 10$.

Simplification of algebraic expressions

When simplifying, follow the order of operations (PEMDAS):

1. Parentheses
2. Exponents
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Here's how the sign rules come into play. Take this expression:

$-3(2x - 5) + 4(-2x + 3)$

1. Distribute $-3$ across $(2x - 5)$: $(-3)(2x) = -6x$ and $(-3)(-5) = +15$

2. Distribute $4$ across $(-2x + 3)$: $(4)(-2x) = -8x$ and $(4)(3) = +12$

3. Combine: $-6x + 15 - 8x + 12$

4. Combine like terms: $-6x - 8x = -14x$ and $15 + 12 = 27$

5. Result: $-14x + 27$

Like terms are terms with the same variable raised to the same exponent. For instance, $3x^2$ and $-5x^2$ are like terms, but $3x^2$ and $3x$ are not.

Substitution in variable expressions

To evaluate an expression, replace each variable with its given value, then simplify using the order of operations.

Example 1: If $x = -2$ and $y = 3$, evaluate $2x - 3y$.

1. Substitute: $2(-2) - 3(3)$

2. Multiply: $-4 - 9$

3. Subtract: $-13$

Example 2: If $a = -3$ and $b = 2$, evaluate $-2ab$.

1. Substitute: $-2(-3)(2)$
2. Multiply left to right: $(-2)(-3) = 6$, then $6 \times 2 = 12$
3. Result: $12$

Notice in Example 2 that you're multiplying three numbers. A quick shortcut: count the negative signs. An even number of negatives gives a positive result; an odd number gives a negative result. Here there are two negatives ($-2$ and $-3$), so the product is positive.

Applying Integer Operations to Word Problems and Real-World Scenarios

Integer word problem conversion

Turning a word problem into math takes three steps:

1. Identify the given information and what you need to find.
2. Assign a variable to the unknown quantity.
3. Translate the words into a mathematical expression or equation.

Example: "John has $20. He owes his friend $35. How much more money does John need?"

• Given: John has $20, owes $35
• Unknown: the additional amount John needs
• Expression: $x = 35 - 20 = 15$, so John needs $15 more

Integer operations in real-world scenarios

Positive and negative integers naturally represent opposite directions or categories. Here are three common contexts:

Temperature changes (positive = increase, negative = decrease)

• "The temperature was 5°C in the morning and dropped by 8°C by evening."
• $T_{\text{evening}} = 5 + (-8) = -3$°C
• The evening temperature was $-3$°C.

Profit and loss (positive = profit, negative = loss)

• "A company made $10,000 in January but lost $5,000 in February."
• $\text{Net profit} = 10{,}000 + (-5{,}000) = 5{,}000$
• The company's net profit over both months was $5,000.

Elevation (positive = above sea level, negative = below sea level)

• "A submarine goes from 100 meters above sea level to 50 meters below sea level."
• $\text{Change in elevation} = -50 - 100 = -150$ meters
• The submarine descended 150 meters total.

In each case, the process is the same: translate the situation into an expression using positive and negative integers, apply the sign rules, then interpret your answer back in context. That last step matters. A result of $-3$ isn't just a number; it means 3 degrees below zero.