Problem 42 - College Algebra Essentials

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 \(x = 1\), given the restriction that \(x ≠ 2\).

Step 1: Identifying Restrictions

The equation given is \(\frac{2}{x-2}=\frac{x}{x-2}-2\). The denominator for both fractions is \(x-2\). Setting this equal to zero gives us \(x-2 = 0\), or \(x = 2\). Thus, \(2\) is the restriction on the variable \(x\). This is because plugging \(x=2\) into the equation would yield a denominator of zero, causing undefined behaviour.

Step 2: Simplify Equation

Given that \(x ≠ 2\), the equation can be simplified. If we multiply each term by \(x-2\), the equation becomes: \(2 = x - 2(x-2)\). This equation is now free of fractions and easier to solve.

Step 3: Solve for the Variable

Let's expand and simplify the equation: \(2 = x - 2x + 4\), which simplifies to \(2x = 2\). Excluding restriction, \(x = 1\), after dividing through by \(2\). However, this value does not violate the restriction obtained in step 1.

Key Concepts

Variables in Denominators

Understanding rational equations requires familiarity with variables in denominators. These are equations where a variable appears under a fraction, acting as the denominator. The fundamental rule when dealing with such equations is to remember that the denominator cannot be zero because division by zero is undefined.
For example, consider the equation \[\frac{2}{x-2} = \frac{x}{x-2} - 2\]Here, the term \(x-2\) is the denominator. As a key rule, you must identify when the denominator equals zero, because these are your restrictions. Whenever a variable is part of the denominator, your first step is to set the denominator equal to zero and solve for the variable.
This will highlight any 'problematic' values that could invalidate the equation application.

Restrictions on Variables

In rational equations, restrictions on variables are values that make any denominator in the equation equal to zero. These restrictions prevent the denominator from becoming zero, thus avoiding undefined mathematical scenarios.
To find the restrictions in our example equation \(\frac{2}{x-2} = \frac{x}{x-2} - 2\), set the denominator \(x-2\) to zero: \[x - 2 = 0\]Solving gives \(x = 2\). This means \(x = 2\) is a restricted value, meaning it cannot be used in any solution because it would result in division by zero.
It is crucial to note these restrictions up-front, as they guide which potential solutions are valid. Always double-check that your final answer does not include any of these restricted values.

Solving Equations with Restrictions

Once you've identified restrictions, solving the rational equation involves simplifying and working through the equation. It helps to eliminate the fractions by multiplying through by a factor that contains the denominator(s), thus simplifying the process.
For the equation \(\frac{2}{x-2} = \frac{x}{x-2} - 2\), after establishing the restriction \(x eq 2\), you can multiply every term by \(x-2\). This way, you eliminate the denominators: \[2 = x - 2(x - 2)\] Next, simplify the equation by distributing the \(-2\) and combining like terms: \[2 = x - 2x + 4\]Keep simplifying to \(-x + 4\), then solve for \(x\). You deduce \(2x = 2\), leading to \(x = 1\).
Finally, verify that this solution does not conflict with any restrictions. Here, \(x = 1\) is not the restricted value \(x = 2\), so it stands as a valid solution.