Problem 5 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. The product or quotient of two numbers with like signs is _____. b. The product or quotient of two numbers with unlike signs is _____.

Step-by-Step Solution

Answer

a. Positive b. Negative

Step 1: Identifying like signs

Like signs mean both numbers are either positive or negative. For example, (+3) and (+5) or (-3) and (-5) are pairs with like signs.

Step 2: Understanding product and quotients with like signs

When multiplying or dividing two numbers with like signs, both positive or both negative, the result is always positive. For example, multiplying two positive numbers (e.g., 3 × 5) results in a positive number 15, and the same applies to negatives (e.g., -3 × -5 = 15).

Step 3: Identifying unlike signs

Unlike signs mean one number is positive and the other is negative. Examples include (+3) and (-5) or (-3) and (+5).

Step 4: Understanding product and quotients with unlike signs

The product or quotient of two numbers with unlike signs is always negative. For example, multiplying a positive number with a negative number (e.g., 3 × -5) yields -15, and the same applies to division (e.g., 15 ÷ -3 = -5).

Key Concepts

Understanding Like Signs

In mathematics, numbers can have either a positive or negative sign. When we say numbers have "like signs," we mean they are either both positive or both negative. This is an important distinction because it affects the outcome when you multiply or divide them.

• For positive like signs: Imagine two positive numbers, like 3 and 5. When you multiply them (3 × 5), you get a positive 15. The rule here is simple: positive multiplied by positive equals positive.


• For negative like signs: Consider negative numbers such as -3 and -5. Multiplying these gives you (−3 × −5) a positive 15, which might seem a bit tricky at first. But think of it like this: "two negatives make a positive." It's the same logic when dividing.

When dealing with like signs, multiply or divide, and you’ll always end up with a positive result. This rule is essential for solving equations properly.

Exploring Unlike Signs

Unlike signs refer to one number being positive and the other negative. This scenario changes the result significantly when performing multiplication or division.

• Imagine you have a positive number, like 3, and a negative number, such as -5. When multiplying these (3 × -5), the result is -15. Notice how the product is a negative number.


• The outcome is the same if you change orders, such as (-3) × 5. You still end up with a negative product, -15.

The principle here is straightforward:

• Multiplying or dividing numbers with unlike signs always results in a negative number.

This idea holds true no matter the magnitude of the numbers involved. Remembering this rule helps in solving problems accurately where differing signs are concerned.

Positive and Negative Numbers

Positive and negative numbers are fundamental in mathematics. Understanding how they interact makes problem-solving easier.

• Positive Numbers: These are numbers without a minus sign. They represent values greater than zero, like 3, 15, or 1000. In terms of visualizing, think of them as positions to the right on a number line.


• Negative Numbers: These have a minus sign before them. Examples are -3 or -15. They indicate values less than zero, often thought of as steps to the left on a number line.

When combining these, the sign of the result often depends on the operations (multiplication or division) and the signs involved.

• Two positive or two negative numbers will always combine to give a positive result during multiplication or division.


• One positive and one negative will always give a negative result.

Grasping these interactions aids comprehension in various mathematical tasks, such as algebra, calculus, and beyond.