Problem 97 - Intermediate Algebra

Question

In the following exercises, multiply or divide. (a) \(-28 \div 7\) (b) \(-180 \div 15\) (c) 3(-13) (d) \(-1(-14)\)

Step-by-Step Solution

Answer

(a) \-4\, (b) \-12\, (c) \-39\, (d) \14

Step 1 - Divide \(-28 \div 7\)

To solve \(-28 \div 7\), divide \text{the absolute values and assign the negative sign}\.: \(-28 \div 7 = -4\)

Step 2 - Divide \(-180 \div 15\)

To solve \(-180 \div 15\), divide the absolute values and assign the negative sign: \(-180 \div 15 = -12\)

Step 3 - Multiply \(3(-13)\)

To solve \(3(-13)\), multiply the absolute values and assign the negative sign: \(3(-13) = -39\)

Step 4 - Multiply \(-1(-14)\)

To solve \(-1(-14)\), multiply the absolute values. Since both are negative, the result is positive: \(-1(-14) = 14\)

Key Concepts

Integer Multiplication

Understanding integer multiplication is crucial when solving problems involving positive and negative numbers.
When multiplying integers, follow these simple rules:

• If you multiply two positive integers, the result is positive (e.g., 3 * 4 = 12).
• If you multiply two negative integers, the result is also positive because the negatives cancel each other out (e.g., \(-3 \times -4 = 12\)).
• If you multiply a positive integer by a negative integer, the result is negative (e.g., 3 * \(-4\) = \(-12\)).

For example, in the exercise, to solve 3 \(-13\), multiply the absolute values 3 and 13 to get 39.
Then, because one number is negative and one is positive, the result is negative: 3 \(-13\) = \(-39\).
Another problem in the exercise is \(-1\) \(-14\). When we multiply these, both numbers are negative, making the result positive: \(-1\) \(-14\) = 14.
Practice makes perfect, so try to multiply more pairs of integers following these rules.

Integer Division

Integer division follows similar rules to multiplication but involves dividing instead.
When dividing integers, remember these key points:

• If you divide two positive integers, the result is positive (e.g., 12 \div 4 = 3).
• If you divide two negative integers, the result is positive (e.g., \(-12\) \div \(-4\) = 3).
• If you divide a positive integer by a negative integer, the result is negative (e.g., 12 \div \(-4\) = \(-3\)).
• If you divide a negative integer by a positive integer, the result is negative (e.g., \(-12\) \div 4 = \(-3\)).

Let's break down the exercise examples:
For \(-28\) \div 7, we divide the absolute values 28 by 7 to get 4.
Since the original integer was negative, the result is negative: \(-28\) \div 7 = \(-4\)
Similarly, for \(-180\) \div 15, divide the absolute values to find 12.
The negative sign of \(-180\) means the result is negative: \(-180\) \div 15 = \(-12\).

Negative Integers

Negative integers can sometimes be tricky, but don't worry—it's simpler than it seems.
A negative integer is simply a number less than zero, denoted with a \- sign.
Understanding how negative integers interact with each other and with positive integers is key:

• When you add two negative integers, you get a larger negative number (e.g., \(-3\) + \(-5\) = \(-8\)).
• When you subtract a negative integer, it is like adding a positive integer (e.g., \(-3\) - \(-5\) = 2).
• When you multiply or divide with negative integers, remember the rules we covered above.

In the exercises, we see negative integers in both multiplication and division.
For example, \(-28\) \div 7 and 3 \(-13\) use the rules of negative integer interactions.
Keep practicing with these tips, and negative integers will soon become second nature.