Problem 51 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ 14 \div(-2) $$

Step-by-Step Solution

Answer

The quotient is -7.

Step 1: Identify the Numbers

In this division problem, identify the dividend and the divisor. The dividend is 14 and the divisor is -2.

Step 2: Divide the Numbers

Divide the numbers as you normally would. Compute \( 14 \div (-2) \).

Step 3: Simplify the Result

When dividing a positive number by a negative number, the result is negative. Hence, \(14 \div (-2) = -7\).

Step 4: Check the Result

To verify, multiply the quotient by the divisor and check if the product equals the dividend. \(-7 \times (-2) = 14\). Since this is true, our division is correct.

Key Concepts

division of integers

Understanding the division of integers is essential for solving many math problems. When dealing with integers, these numbers can be both positive and negative.

Here are the key steps:

• Identify the dividend (the number being divided) and the divisor (the number you are dividing by).
• Conduct the division as you would with positive numbers.
• Consider the sign of both the dividend and the divisor to determine the sign of the quotient.

For example, in the problem \( 14 \div (-2) \), we identify 14 as the dividend and -2 as the divisor. We then conduct the division 14 / 2 as usual to get 7. Because one number is positive and the other negative, the quotient will be negative. Therefore, \( 14 \div (-2) = -7 \).

positive and negative numbers

When dividing integers, it's crucial to understand how positive and negative numbers interact.

General rules include:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative

In our example, 14 is positive, and -2 is negative. Following the rules, a positive divided by a negative equals a negative result. Hence, \( 14 \div (-2) = -7 \).

Having these rules at your fingertips can simplify the process of working with integers.

verifying division results

After computing a division problem, it's essential to verify your result to ensure its accuracy.

Checking the result involves:

• Multiplying the quotient by the divisor
• Confirming that the product equals the original dividend

Following our example, we calculated the quotient to be -7. To check this, we multiply -7 by -2: \[ -7 \times (-2) = 14 \]
The product is 14, which matches our original dividend. This shows our division is correct. Verifying results helps avoid mistakes and ensures confidence in your answers.