Lesson 1.4: Multiply and Divide Integers

New Concept

Master multiplying and dividing integers using one simple rule: same signs yield a positive result, while different signs yield a negative. This skill is key to simplifying expressions, evaluating variables, and solving real-world application problems.

What’s next

Next, you'll apply these rules through interactive examples and practice cards, building your confidence to solve more complex algebraic problems.

Property

For multiplication of two signed numbers:

• If the signs are the same, the product is positive.
• If the signs are different, the product is negative.

Same Signs (Product is Positive)

• Two positives: $7 \cdot 4 = 28$
• Two negatives: $-8(-6) = 48$

Different Signs (Product is Negative)

• Positive $\cdot$ negative: $7(-9) = -63$
• Negative $\cdot$ positive: $-5 \cdot 10 = -50$

Property

Division is the inverse operation of multiplication. For division of two signed numbers, the rules are the same as for multiplication:

• If the signs are the same, the result is positive.
• If the signs are different, the result is negative.

Examples

• To divide $-45 \div 9$, the signs are different, so the quotient is negative. The result is $-5$.
• To divide $-72 \div (-8)$, the signs are the same, so the quotient is positive. The result is $9$.
• To divide $100 \div (-25)$, the signs are different, so the quotient is negative. The result is $-4$.

Explanation

Division uses the exact same sign rules as multiplication. If you know that a negative times a negative is a positive, then you also know that a negative divided by a negative is a positive. The rules are consistent!

Property

The order of operations (PEMDAS/BODMAS) still applies when integers are included. When simplifying expressions, perform operations in the correct sequence: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Be aware of the difference with exponents:

• $(-a)^n$ means the base is $-a$.
• $-a^n$ means the base is $a$, and you find the opposite of the result.

Examples

• To simplify $15 - 4(2 - 9)$, first solve the parenthesis: $15 - 4(-7)$. Then multiply: $15 - (-28)$. Finally, subtract: $15 + 28 = 43$.
• To simplify $(-5)^2 - 30 \div (10-4)$, handle the exponent and parenthesis first: $25 - 30 \div 6$. Then divide: $25 - 5 = 20$.
• To simplify $9(-4) \div (-2)^3$, first evaluate the exponent: $9(-4) \div (-8)$. Then multiply: $-36 \div (-8) = 4.5$.

Property

To evaluate a variable expression means to substitute a number for the variable in the expression and then simplify the resulting expression. When substituting negative numbers, it is often helpful to use parentheses to ensure the operations are performed correctly.

Examples

• To evaluate $3x^2 + 5x - 1$ when $x = -2$, substitute to get $3(-2)^2 + 5(-2) - 1$. This simplifies to $3(4) - 10 - 1 = 12 - 10 - 1 = 1$.
• To evaluate $(a+b)^2$ when $a = -10$ and $b = 7$, substitute to get $(-10+7)^2$. This simplifies to $(-3)^2 = 9$.
• To evaluate $50 - y$ when $y = -15$, substitute to get $50 - (-15)$. This simplifies to $50 + 15 = 65$.

Explanation

Evaluating an expression is like plugging a specific value into a template. The variable is a placeholder, and you replace it with the given number. Using parentheses, especially for negative numbers, prevents mistakes with signs and exponents.

Property

Translating English phrases to algebra involves recognizing keywords for operations. Key words for multiplication and division include:

• Product: Indicates multiplication (e.g., 'the product of $a$ and $b$' is $a \cdot b$).
• Quotient: Indicates division (e.g., 'the quotient of $a$ and $b$' is $a \div b$ or $\frac{a}{b}$).

Be careful with subtraction order: 'the difference of $a$ and $b$' is $a-b$, but '$b$ subtracted from $a$' is also $a-b$.

Examples

• The phrase 'the product of $-8$ and $12$' translates to the expression $(-8)(12)$, which simplifies to $-96$.
• The phrase 'the quotient of $-100$ and $-4$' translates to the expression $-100 \div (-4)$, which simplifies to $25$.
• The phrase 'the sum of $-5$ and $3$, increased by $10$' translates to $[-5 + 3] + 10$, which simplifies to $-2 + 10 = 8$.

Explanation

This skill is like translating from one language to another. Words like 'product' and 'quotient' are direct signals for multiplication and division. Reading carefully helps put the numbers and operations in the correct order to match the phrase.

Property

To solve applications with integers, use a structured approach:

1. Read the problem to understand all words and ideas.
2. Identify what you are asked to find.
3. Write a phrase that gives the information to find it.
4. Translate the phrase into a mathematical expression.
5. Simplify the expression.
6. Answer the question with a complete sentence.

Examples

• A scuba diver descends 20 feet per minute for 5 minutes. To find her change in depth, translate '5 times a 20-foot descent' to $5(-20)$. The change is $-100$ feet.
• A company had a total loss of 6,000 dollars over 4 months. To find the average monthly loss, translate 'the quotient of $-6000$ and $4$' to $-6000 \div 4$. The average loss was 1,500 dollars per month.
• The temperature at noon was $5^\circ$C. By midnight, it dropped to $-11^\circ$C. The difference is $5 - (-11) = 16^\circ$C.

Explanation

This six-step plan turns a word problem into a straightforward math problem. It helps you organize your thoughts, translate the situation into an expression, solve it, and then state the answer clearly in the context of the original problem.