Problem 42 - Algebra and Trigonometry

Question

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$\frac{2}{x-2}=\frac{x}{x-2}-2$$

Step-by-Step Solution

Answer

The solution to the equation is \(x=4\).

Step 1: Identify the Restrictions

Equating the denominator to 0 to find the values that make the denominator zero i.e., \(x-2=0\). Solving this gives \(x=2\). So, \(x=2\) is a restriction.

Step 2: Solve the Equation

Next step is to solve for x considering the restrictions. Simplifying the equation \(\frac{2}{x-2}=\frac{x}{x-2}-2\), we find that \(2=x-2\). Adding 2 to both sides gives \(x=4\).

Step 3: Check for Restrictions

Checking our solution against our restrictions, we find that \(x=4\) does not violate the restriction \(x≠2\). Thus, \(x=4\) is the solution to the equation.

Key Concepts

Variable Restrictions

When solving rational equations, it is crucial to identify variable restrictions first. Variable restrictions occur when the equation might become undefined, typically due to a zero in the denominator. These restrictions must be checked before proceeding with the solution.

Here's how to handle variable restrictions:

• Identify denominators in the equation. For example, in \( \frac{2}{x-2} \), the denominator is \( x-2 \).
• Set the denominator equal to zero to find the restriction. For \( x-2 \), you solve \( x-2 = 0 \), resulting in \( x = 2 \).
• Note these values as restrictions because they cause the equation to be undefined.

By understanding and identifying these restrictions first, you avoid pursuing solutions that don't make sense in the problem's context.

Denominator Zero

Understanding why a denominator cannot be zero is essential in mathematics. Fractions become undefined when their denominator equals zero, thus we must ensure no such values exist within our solutions.

Consider why a zero denominator is problematic:

• Mathematically, dividing by zero is undefined. For example, \( \frac{1}{0} \) doesn't have a meaningful value in the real world.
• Equations are only valid for values that make sense logically. Hence, checking denominators is necessary to avoid errors and misconceptions.

In practice, always check all denominators by setting them to zero and solving. The resulting values are the points at which the equation is undefined, highlighting restrictions and ensuring solutions are logical and valid.

Solving Equations

Once restrictions are identified, the next step is solving the equations mindfully considering those restrictions. Rational equations often involve finding a common denominator or simplifying the equation correctly.

Tips for solving rational equations:

• Start by simplifying both sides of the equation as much as possible. For instance, \( \frac{2}{x-2} = \frac{x}{x-2} - 2 \).
• Add or subtract terms wisely to isolate the variable. Continue solving by combining like terms, as in converting \( \frac{2}{x-2} = \frac{x-2}{x-2} \).
• Check that any solution obtained doesn't violate the earlier identified restrictions. For example, \( x = 4 \) doesn’t conflict with \( x = 2 \).

By following these logical steps and double-checking against restrictions, you'll effectively solve rational equations and deepen your understanding.