Problem 42 - College Algebra

Question

Exercises \(31-50\) contain 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

No solution because the potential solution \(x = 2\) causes the fraction to have a denominator of 0, making it undefined.

Step 1: Identify the Restrictions

Solving the equation \(x - 2 = 0\) for \(x\) shows that \(x = 2\) is the restriction. This is because it would make the denominator of a fraction 0, which is undefined in mathematics.

Step 2: Rearrange the Equation

Given that \(x\neq2\), the denominators on both sides will not be zero. So, we can eliminate them by multiplying every term by \(x-2\). The equation becomes \(2 = x - 2(x-2)\).

Step 3: Solve the Equation

Solve the equation for \(x\). Start by expanding the right-hand side to get \(2 = x - 2x + 4\). Combining like terms yields \(2 = -x + 4\). Solving this for \(x\) finally gives us \(x = 2\).

Step 4: Check the Solution

Even though algebraically \(x = 2\) is the solution, \(x = 2\) was already defined as the restriction of the original equation. Hence, the original equation has no solution because the only potential solution is outside the domain of the equation.

Key Concepts

Variable Restrictions

When working with rational equations, it's important to identify any restrictions on the variables. These restrictions occur because certain variable values can make the denominator of a fraction equal to zero, leading to an undefined expression. In the given problem, the equation is \[\frac{2}{x-2}=\frac{x}{x-2}-2\]. We must ensure that the denominator, \(x-2\), is not zero, as this would make the expression undefined.

To find where the denominator equals zero, we solve \(x-2=0\). This calculation shows that \(x=2\) is a restriction because it results in a zero denominator. Restrictions are crucial because they define the domain of the equation, meaning any solution must adhere to these limitations. When solving equations with fractions, always remember to find and consider variable restrictions.

Denominators

The denominator in a rational equation is the number or expression located below the fraction line. It plays a crucial role in determining the equation's restrictions and domain. In our problem, the expression \(x-2\) serves as the denominator for fractions on both sides of the equation.

Identifying denominators helps to determine the restrictions that keep the equation valid. If a denominator becomes zero for any value of the variable, mathematical operations cannot proceed, as division by zero is undefined.

• Always find where the denominator equals zero.
• Use this value to establish variable restrictions.
• Ensure all potential solutions respect these restrictions.

Understanding denominators in rational equations is key to successfully solving them, as they influence both solution methods and validity.

Solving Equations

Once restrictions are identified, solving the rational equation becomes straightforward. With our equation, eliminating the denominators simplifies the process. We start with: \[\frac{2}{x-2}=\frac{x}{x-2}-2\].

Knowing that \(x eq 2\), we safely multiply through by \(x-2\) to clear the denominators, obtaining:\[2 = x - 2(x-2)\].
Expanding and simplifying the equation gives:\[2 = x - 2x + 4\], which reduces to \[2 = -x + 4\]. By solving for \(x\), we find \[x = 2\]. However, this value disobeys the restrictions, rendering the equation unsolvable within its valid domain. Always verify that obtained solutions abide by the initial conditions of the problem.

Undefined in Mathematics

In mathematics, expressions become undefined when they involve operations that are not allowed. Division by zero is one such operation. It is important to identify conditions under which parts of an expression cannot be evaluated, to prevent undefined results.

In our equation \(\frac{2}{x-2}=\frac{x}{x-2}-2\), the restriction \(x=2\) leads to a zero denominator, resulting in an undefined fraction. This means that while it may seem like \(x=2\) might solve the equation, it is actually not viable because it produces an undefined expression.

• Undefined expressions invalidate solutions.
• Always check solutions against identified constraints.
• Ensure expressions remain defined for all valid solutions.

Mathematical expressions must be carefully assessed to avoid undefined elements, ensuring the integrity of the solutions and operations performed.