Problem 49 - Elementary Algebra

Question

Explain why the quotient \(\frac{x-2}{x+1} \div \frac{x}{x-1}\) is undefined for \(x=-1, x=1\), and \(x=0\) but is defined for \(x=2 .\)

Step-by-Step Solution

Answer

The quotient is undefined for \(x=-1\), \(x=1\), and \(x=0\) due to division by zero, but defined for \(x=2\).

Step 1: Understand Division of Fractions

The expression discussed is a division of two fractions, \(\frac{x-2}{x+1} \div \frac{x}{x-1}\). Division of fractions can be transformed into multiplication by flipping the second fraction. This gives us: \(\frac{x-2}{x+1} \times \frac{x-1}{x}\).

Step 2: Identify Critical Points

The critical points occur when the denominator equals zero. Thus, we need to solve \(x+1=0\), \(x-1=0\), and \(x=0\). This gives critical values of \(x=-1\), \(x=1\), and \(x=0\).

Step 3: Evaluate Each Critical Point

For \(x=-1\), the first fraction becomes undefined as \(x+1 = 0\). For \(x=0\), the second fraction after flipping becomes undefined. For \(x=1\), the second fraction is undefined since \(x-1 = 0\). All these points make the original expression undefined.

Step 4: Check If Defined at x=2

Substituting \(x=2\) into the expression, both denominators \(x+1\) and \(x-1\) are non-zero, and \(xeq 0\). Hence, the expression is defined for \(x=2\).

Key Concepts

Division of Fractions

Understanding how to divide fractions is essential when tackling algebraic expressions. It might seem tricky at first, but with a simple rule, it becomes easy to manage. Turning division into multiplication is the key. To divide by a fraction, you multiply by its reciprocal. In simple terms, flipping the second fraction will change the division into multiplication.
In mathematical notation, if you have two fractions, \(\frac{a}{b} \div \frac{c}{d}\), write it as \(\frac{a}{b} \times \frac{d}{c}\). This adjustment can simplify many problems significantly.

• First, flip the second fraction.
• Second, change the ÷ to ×.
• Lastly, multiply the fractions.

So, our expression \(\frac{x-2}{x+1} \div \frac{x}{x-1}\) becomes \(\frac{x-2}{x+1} \times \frac{x-1}{x}\). With this form, it’s easier to spot potential issues like undefined points.

Undefined Expressions

In algebra, expressions become undefined when the denominator of a fraction is zero. It is impossible to divide by zero because division by zero does not yield a meaningful result. Keep this in mind as a major rule of algebraic operations. To see if and when an expression is undefined, check its denominator. If it ever equals zero, the expression cannot be evaluated.
For the expression \(\frac{x-2}{x+1} \times \frac{x-1}{x}\), calculate when each denominator equals zero:

• For \(x+1\), it's zero when \(x = -1\).
• For \(x-1\), it's zero when \(x = 1\).
• For \(x\), it's zero when \(x = 0\).

This exposes the critical points: \(x = -1\), \(x = 1\), and \(x = 0\) make the expression undefined.

Critical Points in Algebra

Critical points in algebra are values that cause a fraction's denominator to be zero, rendering the expression undefined. Detecting these points is crucial when solving algebraic problems. It deals with keeping the mathematics sound and meaningful by ensuring no operations involve division by zero.
In the provided expression \(\frac{x-2}{x+1} \times \frac{x-1}{x}\), we've identified the critical points as \(x = -1\), \(x = 1\), and \(x = 0\). Each point corresponds to a denominator equaling zero, explaining why these values make the expression problematic.

• \(x = -1\) turns \(x+1\) to zero.
• \(x = 1\) turns \(x-1\) to zero.
• \(x = 0\) turns \(x\) to zero.

By spotting and understanding these points, you can sidestep potential errors and verify solutions more efficiently. However, note that at \(x = 2\), none of the denominators are zero, and thus the expression is defined.