Problem 69 - Elementary and Intermediate Algebra

Question

Add. $$ -20+(-16)+10 $$

Step-by-Step Solution

Answer

The sum is -26.

Step 1: Add the First Two Numbers

Begin by adding the first two numbers: \(-20 + (-16)\).Since both numbers are negative, you add their absolute values and keep the negative sign:\(|-20| + |-16| = 20 + 16 = 36\).Thus, \(-20 + (-16) = -36\).

Step 2: Add the Result to the Third Number

Now, add the result from Step 1 to the third number:\(-36 + 10\).Since one number is negative and the other is positive, subtract the smaller absolute value from the larger:\(|-36| - |10| = 36 - 10 = 26\).Since the number with the larger absolute value is negative, the result will also be negative:\(-36 + 10 = -26\).

Key Concepts

Understanding Absolute Value

The absolute value of a number refers to its distance from zero on the number line, regardless of its direction. Essentially, absolute value strips away any negative sign, leaving just the magnitude.
For example, the absolute value of

• -20 is 20, because it is 20 units away from zero on the number line.
• Similarly, |-16| equals 16.

Absolute value is important for operations involving negative numbers because it helps in determining the size of numbers without considering their sign. We'll utilize absolute values in addition rules to simplify calculations by initially ignoring the signs.

Working with Negative Numbers

Negative numbers are the numbers located to the left of zero on the number line, and they are represented with a minus sign (-). They are quite the opposite of positive numbers and when dealing with them, special rules apply.
When adding or subtracting, carefully consider the signs of the numbers involved:

• Adding two negative numbers results in a more negative number because you are essentially increasing the distance in the negative direction. For example, \(-20 + (-16) = -36\).
• When performing mixed additions (a negative number plus a positive number), subtract the smaller absolute value from the larger. The result will take the sign of the number with the larger absolute value, just like \(-36 + 10 = -26\).

Handling negative numbers correctly requires practice and a solid understanding of these principles.

Adding Integers Using Addition Rules

Adding integers involves several rules that simplify the process. Mastering these rules helps in solving problems quickly and accurately:

• Same Sign Addition: When both numbers have the same sign (both positive or both negative), add their absolute values and keep the common sign. This was used when adding \(-20 + (-16)\).
• Different Sign Addition: If the numbers have different signs, subtract the smaller absolute value from the larger and apply the sign of the number with the larger absolute value. This was applied to \(-36 + 10\).

These addition rules are crucial in performing basic arithmetic with integers and forming a strong numerical foundation, especially for dealing with more complex expressions.