Problem 52 - Fundamentals of Mathematics

Question

Determine each value. $$ |-7|-|-10| $$

Step-by-Step Solution

Answer

The value is \(-3\).

Step 1: Understand the Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. In other words, it changes any negative signs to positive ones. Therefore, \(|-7| = 7\) and \(|-10| = 10\).

Step 2: Substitute Values into the Expression

Substitute the values we found from the absolute values into the given expression. This gives us: \[ 7 - 10 \]

Step 3: Solve the Expression

Perform the subtraction: \[ 7 - 10 = -3 \]

Key Concepts

Number Line

The number line is a visual tool used in mathematics to represent integers and their order. It's essentially a straight line with numbers placed at equal intervals along its length. - **Zero** is the center, splitting positive numbers (to the right) and negative numbers (to the left). For example, on a number line, -7 is located to the left of zero, while 7 is on the right. The number line helps us understand concepts like the absolute value and distance from zero by providing a simple graphical representation.

Subtraction

Subtraction is a basic arithmetic operation that determines the difference between two numbers. It involves taking one number away from another.- In the expression \(7 - 10\), 7 is the minuend (the number from which another number is subtracted) and 10 is the subtrahend (the number to be subtracted).- The result of this operation gives us the difference. Since 7 is less than 10, the result is a negative number, -3, meaning the difference places us three units left of zero on the number line.Understanding subtraction helps us determine the relationship in size or value between two numbers.

Distance from Zero

The distance from zero on a number line is what defines the absolute value of a number. - Absolute value is always non-negative, as it measures only how far a number is from zero, ignoring direction. - For instance, both -7 and 7 have an absolute value of 7, because they are each seven units away from zero, just in opposite directions. This concept is crucial when solving problems with absolute values because it helps us identify the actual size of a number without worrying about its sign.