Problem 32 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ 144+(-88) $$

Step-by-Step Solution

Answer

The result is 56.

Step 1: Identify the Operation

The exercise presents a basic arithmetic operation which is the addition of two numbers: 144 and -88.

Step 2: Consider the Signs

Notice that 144 is positive and -88 is negative. We are adding a positive number and a negative number, which essentially means performing a subtraction.

Step 3: Subtract the Smaller Absolute Value

Subtract the smaller absolute value from the larger one: 144 - 88. Calculate this difference to get 56.

Step 4: Determine the Sign of the Result

Since the absolute value of 144 is larger than that of 88, the result will take the sign of 144, which is positive. Therefore, the result is +56.

Key Concepts

Integer Operations

When we deal with integer operations, the most common are addition and subtraction. While it might seem straightforward, combining positive and negative numbers requires a bit more understanding.

With addition of integers, it's crucial to consider their signs:

• If both integers have the same sign, you simply add the absolute values and keep the same sign.
• If the integers have different signs, like in the problem we are examining, you subtract the absolute values and the result takes the sign of the integer with the larger absolute value.

These rules ensure that we correctly balance positive and negative influences when they interact in an equation. This concept is essential, especially when dealing with equations in algebra and more advanced math.

Absolute Value

Understanding absolute value can greatly simplify working with integers. The absolute value of a number is the distance from zero on a number line, regardless of direction—it is always non-negative. It tells us how 'big' or 'strong' the integer is, without considering whether it's positive or negative.

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is the positive counterpart of that number.

For example, the absolute values are:

• |144| = 144
• |-88| = 88

When adding integers like 144 and -88, understanding which absolute value is larger helps determine both the operation (addition or subtraction) and the sign of the result, as shown in the steps provided earlier.

Positive and Negative Numbers

Positive and negative numbers form the basis of integer operations. Positive numbers are greater than zero and are typically to the right of zero on a number line. Negative numbers, on the other hand, are less than zero and found to the left. Adding a negative number is akin to subtracting its positive counterpart. This is why understanding the signs is critical in integer operations. For example, adding -88 is like subtracting 88.
When integers with opposite signs interact, the outcome depends on which has a larger absolute value. Hence, in the equation 144 + (-88), since 144, with its positive sign, has a larger absolute value than -88, the result is positive. Grasping this concept helps unravel numerous math problems efficiently.