Problem 8 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ -10+5 $$

Step-by-Step Solution

Answer

The sum of -10 and 5 is -5.

Step 1: Identify the numbers

In the expression \[-10+5\], the numbers involved are \(-10\) and \(+5\). The operator used between these numbers is addition (+).

Step 2: Interpret the expression

\(-10\) indicates a negative ten, and \(+5\) indicates positive five. We'll be adding a positive number to a negative number.

Step 3: Use additive inverse property

In arithmetic, adding a positive number to a negative number involves finding the difference between the absolute values of the two numbers. The absolute value of \(-10\) is \(10\), and the absolute value of \(+5\) is \(5\).

Step 4: Calculate the result

Find the difference: \(10 - 5 = 5\). Since \(-10\) has a larger absolute value than \(5\), the result will take the sign of \(-10\). Therefore, the result is \(-5\).

Step 5: Finalize the result

The final result of adding \(-10 + 5\) is \(-5\). The final arithmetic operation confirms that adding a smaller positive number to a larger negative number results in a negative number.

Key Concepts

Negative Numbers

Understanding negative numbers is crucial when learning how to perform operations like addition and subtraction. Negative numbers are numbers less than zero and are typically represented with a minus sign (-) in front of them. For example,

• -1, -2, -3
• -10, -100
• -0.5, -2.75

Negative numbers can be thought of as moving left on a number line, away from zero.
When you add a positive number to a negative number, you are essentially moving the value closer to zero. Conversely, adding a larger negative number than the positive number means you'll continue to move away from zero. Understanding the direction and magnitude of numbers on the number line helps make sense of operations involving negative numbers.

Absolute Value

The absolute value of a number expresses its distance from zero on the number line, without considering its direction (positive or negative). It is denoted by two vertical bars, for example, \[ |-10| = 10 \] and \[ |5| = 5 \]. Key Features of Absolute Value:

• The absolute value is always a non-negative number.
• \(|x| = x\) if \(x\geq0\), and \(|x| = -x\) if \(x<0\).
• The absolute value disregards whether a number is positive or negative.

Using absolute values in arithmetic can simplify complex calculations by focusing only on the size of the numbers. In our original exercise, it helps establish the size difference between \(-10\) and \(5\), and aids in solving the problem effectively.
This knowledge is useful in numerous math concepts, such as distance, magnitude, and error analysis.

Additive Inverse Property

The additive inverse property involves finding a number that, when added to the given number, results in zero. Every real number has an additive inverse, and together they "cancel out" each other.
An example is that the additive inverse of \(+5\) is \(-5\), since \(5 + (-5) = 0\).
In addition problems, this property can sometimes aid in simplifying calculations or confirming results.
Here are some quick facts:

• The additive inverse for \(a\) is \(-a\).
• The sum of a number and its additive inverse is always zero.
• This property is crucial in solving equations and algebraic expressions.

When performing operations like \(-10 + 5\), instead of adding directly, you can think about the way numbers offset one another, facilitating an easier understanding of why the result is \(-5\).
Together with understanding absolute value and negative numbers, this property ensures you comprehend essential arithmetic operations deeply.