Problem 83 - Algebra 1

Question

In Exercises \(83-85,\) choose the statement that is true about the given numbers. (A) The number in column A is greater. (B) The number in column B is greater. (C) The two numbers are equal. (D) The relationship cannot be determined from the given information. Column A Column B $$ -\left|\frac{2}{3}\right|-\frac{2}{3} | $$

Step-by-Step Solution

Answer

The two numbers are equal. So, the statement C: 'The two numbers are equal' is the correct one.

Step 1: Understanding Absolute Value

We know that absolute value of a number is its distance from zero on the number line, which is always non-negative. This means that the absolute value of -2/3 is the same as the absolute value of 2/3, which equals to 2/3.

Step 2: Calculate the numbers in both columns

Take the negative of the absolute value in column A, we have -|2/3| = -2/3. Whereas in column B, the value remains as it is, which is -2/3.

Step 3: Compare the numbers

We can see that the value in Column A (-2/3) is equal to the value in Column B (-2/3).

Key Concepts

Understanding Absolute Value

The concept of absolute value is often visualized as the "distance" a number is from zero on the number line. It is important to note that this distance is always a non-negative number. For example, both -3 and 3 have an absolute value of 3 because they are both 3 units away from zero. In mathematical terms, for any given number \( x \), the absolute value is denoted by \( |x| \). This is why \(|-2/3| = |2/3| = 2/3\).

When dealing with absolute values, key points to remember are:

• The absolute value of a positive number is simply the number itself.
• The absolute value of zero is zero.
• The absolute value of a negative number is the positive version of that number.

Understanding these principles helps us determine that \( |-2/3| \) changes to \( 2/3 \), highlighting its non-negative nature despite the initial negative sign.

Dealing with Negative Numbers

Negative numbers often represent quantities that are below zero, such as a deficit or loss. When working with negative numbers, their appearance on a number line is crucial - they are always positioned to the left of zero.

To tackle comparisons involving negative numbers:

• Remember that, on the number line, leftmost numbers are always smaller. Hence, -5 is less than -3.
• Adding a negative value is equivalent to subtracting its absolute counterpart, while subtracting a negative is like adding, due to the double negative turning positive.

In our exercise, when the expression \(-|2/3|\) is evaluated as \(-2/3\), it does not exceed or fall short of \(-2/3\) from column B, as both are identical negative numbers.

Principles of Number Comparison

Number comparison is the process of determining which of two numbers is greater, lesser, or if they are equal. This is easily done when both numbers are either positive or negative but needs more caution when dealing with mixed states.

Here are some basics for comparing numbers:

• For two positive numbers, the number with the greater absolute value is larger.
• For two negative numbers, the number with the smaller absolute value (i.e., the number closer to zero) is larger.
• If comparing two numbers where one is positive and one is negative, the positive number is always larger.

In our specific example, both numbers \(-2/3\) from Column A and Column B are equal because they share the same numerical value despite being negative. This exercise illustrates how properly interpreting absolute values and understanding negative numbers can lead to an accurate comparison.