Problem 76 - Algebra A Combined Function

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make each statement true. $$ |-12| \quad \frac{-24}{2} $$

Step-by-Step Solution

Answer

|-12| > \frac{-24}{2}

Step 1: Calculate the Absolute Value

First, we need to calculate the absolute value of \(-12 \). The absolute value of a number is its distance from zero on the number line, regardless of direction. So, \( |-12| = 12 \).

Step 2: Simplify the Fraction

Next, simplify the fraction \( \frac{-24}{2} \). By dividing \(-24\) by \(2\), we get \(-12\).

Step 3: Compare the Two Values

Now, we compare the two calculated values: \( 12 \) from the absolute value, and \(-12\) from the fraction. Since \(12\) is greater than \(-12\), the correct inequality to use is \(>\).

Step 4: Conclusion

Thus, inserting the \(>\) sign, the statement becomes \(|-12| > \frac{-24}{2}\).

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions used to compare two values or expressions to see which is larger, smaller, or equal. In these expressions, symbols like \(<\), \(>\), and \(=\) are used to denote these relationships. When comparing two numbers, as in this exercise, you determine whether one number is greater than, less than, or equal to the other.

To correctly establish these relationships, it's crucial to fully compute or simplify each part of the expression being compared. For example, in this exercise, the absolute value must be calculated, and the fraction simplified, before an accurate comparison can be made.

• When \(a > b\), it means 'a' is greater than 'b'.
• When \(a < b\), it means 'a' is less than 'b'.
• When \(a = b\), the values are equal.

Simplifying Fractions

Simplifying fractions is a process of reducing a fraction to its simplest form to make calculations and comparisons easier. This process often involves dividing both the numerator and the denominator by their greatest common divisor, though in some exercises, like this one, it can include directly performing the division.

In the example from the exercise, we started with \(-\frac{24}{2}\). By performing the division, \(-24 \div 2\), we obtain \(-12\). Here, the fraction has been simplified to a whole number, facilitating a straightforward comparison.

When simplifying fractions:

• Divide the top and bottom of the fraction by the same number (they must be factors of both numbers).
• Continue simplifying until no further reduction is possible.
• A simplified fraction has no common factors other than 1.

Number Line Concepts and Absolute Values

The number line is a visual representation of numbers in a straight line where each point corresponds to a number. It extends infinitely in both directions with zero positioned at the center. Understanding how to navigate this line helps in understanding concepts like absolute value and comparing sizes of numbers.

Absolute value refers to the distance of a number from zero on the number line, ignoring the direction. It gives only the magnitude, not the sign. For instance, the absolute value of both \(-12\) and \(12\) is \(12\) because they are both \(12\) units away from zero.

Key points:

• Numbers to the right are greater, and those to the left are lesser.
• Absolute value is always non-negative.
• Visualizing numbers on a line helps in grasping abstract concepts from a mathematical perspective.