Problem 34 - Elementary Algebra

Question

Find the sums. \(21+(-4)\)

Step-by-Step Solution

Answer

Answer: The sum of 21 and -4 is 17.

Step 1: Identify the numbers

First, identify the two numbers that need to be added: \(21\) and \((-4)\). One number is positive while the other is negative.

Step 2: Compare the numbers

Determine which number has a greater absolute value. In this case, \(21\) (the positive number) has a greater absolute value than \((-4)\) (the negative number).

Step 3: Subtract absolute values

Next, subtract the absolute value of the smaller number from the absolute value of the larger number. In this case, \(|21| - |-4| = 21 - 4 = 17\).

Step 4: Determine the sign

Finally, the sign of the resulting sum will be the same as the sign of the number with the greater absolute value. So, the sign of the sum will be positive.

Step 5: Write the final sum

We now know that the sum of \(21\) and \((-4)\) is \(17\) because the calculation is \(21 - 4\). So, the final sum is \(21 + (-4) = 17\).

Key Concepts

Understanding Absolute Value

Absolute value is a concept in mathematics used to describe how far a number is from zero, regardless of the direction on the number line. It's always non-negative. To find the absolute value of a number, you simply remove the sign. For example:
\[ |21| = 21 \]
\[ |-4| = 4 \]

Absolute value helps us compare the size of numbers without worrying about their sign. It's especially useful in integer addition, where determining the larger value can guide us in finding the resulting sum.

Positive and Negative Numbers

Positive and negative numbers represent opposite directions on the number line. Positive numbers are greater than zero and are usually to the right of zero. Negative numbers are less than zero and lie to the left.

The rule of thumb in addition is:

• Adding two positive numbers always gives a positive sum.
• Adding two negative numbers results in a more negative sum.
• When adding a positive and a negative number, the sign of the sum depends on which number has the larger absolute value.

In our exercise, we add a positive \(21\) and a negative \(-4\). Since the positive number is larger in absolute value, the sum will remain positive.

Step by Step Math Solution

Following a structured approach helps avoid confusion, especially in operations like integer addition with mixed signs. A step-by-step method ensures you cover all bases and reach the correct answer:

1. **Identify and list the numbers**:

• We've got two numbers: \(21\) and \(-4\).

2. **Compare their absolute values**:

• Check which number is larger in absolute value. Here, \(21\) is greater than \(|-4|\).

3. **Subtract the smaller absolute value from the larger**:

• Subtract \(4\) from \(21\), leading to \(17\).

4. **Determine the sign of the result**:

• The sign is determined by the larger absolute value, which is positive here.

5. **Write the final result**:

• Thus, \(21 + (-4) = 17\).

By following these steps, you can confidently solve similar problems without getting mixed up.