Problem 15 - Elementary Algebra

Question

Find the sums. 1.5+(-2)

Step-by-Step Solution

Answer

Answer: -0.5

Step 1: Identify the numbers to be added

We have two numbers: 1.5 (positive) and -2 (negative)

Step 2: Add the positive and negative numbers

To add a positive number to a negative number, we need to consider both the rules for adding positive and negative numbers. Since the numbers have different signs, we have to subtract and take the sign of the number with the larger absolute value. To do this, we compare the absolute values of the numbers: |-2| = 2 and |1.5| = 1.5. Since 2 > 1.5, the sum will be the difference between the numbers (2 - 1.5) with the sign of the larger absolute value number, which is -2.

Step 3: Calculate the sum

Now we subtract the numbers and keep the sign of the larger absolute value number: 2 - 1.5 = 0.5. Since -2 has a larger absolute value, the sum will be negative: -0.5 Therefore, the sum of 1.5 and -2 is -0.5.

Key Concepts

Understanding Absolute Values

When working with positive and negative numbers, understanding absolute values is crucial. The absolute value of a number is the distance between that number and zero on the number line, without considering direction. Essentially, it means stripping away the number's sign.

For example, the absolute value of \(-2\) is \(2\) because \(2\) is \(2\) units away from zero. Similarly, the absolute value of \(1.5\) is \(1.5\). Absolute values are always non-negative, which makes them a useful tool for comparing the magnitude of numbers without worrying about whether they are positive or negative.

In our exercise, we compared the absolute values of \(1.5\) and \(-2\) to determine which number has a larger impact on the final sum.

The Role of Number Signs

Number signs indicate whether a number is positive or negative, and they play a vital role in calculations. A positive sign (+) means the number is more than zero, while a negative sign (-) indicates a number less than zero. These signs determine how two numbers will interact when added together.

When adding numbers with different signs, it is essential to understand that you will be combining opposing forces. A positive sign cancels out some portion of a negative sign and vice versa. The result heavily depends on the numbers' absolute values and which sign is associated with the larger number.

• If the positive number is larger, the result will be positive.
• If the negative number is larger, the result will be negative.
• If they are the same, the result is zero.

In the example \(1.5 + (-2)\), the negative number, \(-2\), has a larger absolute value, which causes the final sum to be negative.

Subtraction in Addition Problems

Adding a negative number is often seen as subtracting. In simple terms, \(a + (-b)\) is the same as \(a - b\). This perspective helps simplify understanding and solving addition problems that include negative numbers.

To solve addition problems involving a negative number, consider the direction change on the number line. Instead of moving to the right (which is what happens with addition of positive numbers), move to the left, as if you are subtracting. In our exercise, adding \(-2\) to \(1.5\) translates to moving \(2\) units left from \(1.5\), or subtracting \(1.5\) from \(2\), giving the final sum of \(-0.5\).

• Recast subtraction as adding a negative.
• This makes it easier to apply consistent rules.
• Helps understand direction on the number line.

With this mindset, tackling real-world problems involving both positive and negative numbers becomes more intuitive.