Problem 34 - Elementary Algebra

Question

For the following problems, show that the fractions are equivalent. $$ \frac{-2}{3} \text { and }-\frac{2}{3} $$

Step-by-Step Solution

Answer

Answer: Yes, they are equivalent.

Step 1: Rewrite the first fraction

We need to rewrite the first fraction as the negative of the second fraction. $$ -\frac{2}{3} = -\frac{2}{3} $$

Step 2: Verify their equivalence

Since both expressions are equal, the given fractions are equivalent: $$ \frac{-2}{3} = -\frac{2}{3} $$

Key Concepts

Negative Fractions

Negative fractions are just like regular fractions, but with a minus sign either in front of the entire fraction or with the numerator. An important aspect to remember is that negative fractions represent values less than zero. For instance, \( -\frac{2}{3} \) and \( \frac{-2}{3} \) are both negative fractions. They may look slightly different, but their meaning is identical.

• A negative sign can be placed in front of the whole fraction: \( -\frac{numerator}{denominator} \).


• A negative sign can also be part of the numerator: \( \frac{-numerator}{denominator} \).

Both placements effectively mean the same thing. They indicate that the value is less than zero. The placement doesn't change the fraction's value. When dealing with negative fractions, always focus on the minus sign, as it tells you the fraction is taking away from something, not adding to it.

Fraction Equivalence

Fraction equivalence is a key concept in understanding fractions. Two fractions are considered equivalent if they represent the same part of a whole, even if they look different at first glance. In the case of negative fractions, like \( \frac{-2}{3} \) and \( -\frac{2}{3} \), determining their equivalence involves recognizing that the negative sign does not affect the comparison of their absolute values.

Here’s how you can verify the equivalence of fractions:

• Check if both fractions can be rewritten as identical expressions.


• Convert one fraction into another by mathematically manipulating the negative sign (if necessary).


• Ensure that both numbers have the same numerator and denominator when the negative is uniformly applied.

This method shows how fractions that look different are essentially the same in value. Especially with negative fractions, placing the negative sign at different positions doesn't impact their equivalence since they ultimately reflect the same quantity.

Algebraic Fractions

Algebraic fractions are fractions where the numerator or the denominator contains an algebraic expression. These can often seem complex, but the principles of handling them, including the concept of equivalence and negatives, remain the same as with numeric fractions.

When working with algebraic fractions, it’s essential to apply the same operations you would with simple fractions:

• Simplification by cancelling out common factors.


• Ensuring like terms and coefficients are equivalent when comparing fractions or in operations such as addition and subtraction.


• Manipulating expressions without altering their values and thereby keeping them equivalent.

Understanding how to handle negative signs and ensuring fractions are simplified correctly allows you to compare and equate algebraic fractions effectively. Thus, whether you’re working with numbers or variables, these foundational concepts remain consistent.