For each expression in part (a), perform the indicated...

For each expression in part (a), perform the indicated operations and then simplify, if possible. Solve equation in part (b) and check the result.
a. $\frac{m}{m-2}-\frac{1}{m-3}$
b. $\frac{m}{m-2}-\frac{1}{m-3}=1$

Key Concepts

Solving Rational Equations

When solving equations that include rational expressions, the strategy typically involves eliminating the fractions by multiplying both sides of the equation by the least common denominator. This transforms the equation into a polynomial equation, which can then be solved using standard algebraic techniques.

Simplification of Rational Expressions

Once a common denominator is found and the expressions are rewritten, the numerators are combined and simplified by combining like terms and canceling any common factors. This process makes the expression easier to work with or interpret.

Checking for Extraneous Solutions

After solving a rational equation, it is crucial to substitute the solutions back into the original equation to ensure they do not lead to undefined expressions, such as having a zero in the denominator. This step helps to identify and discard any extraneous solutions.

Rational Expressions

Rational expressions are fractions involving polynomials in the numerator and denominator. They are manipulated similarly to numerical fractions, but additional care must be taken regarding the values of the variable that make any denominator equal to zero.

Common Denominator

Finding a common denominator is essential when adding or subtracting rational expressions. This involves determining a shared denominator, often the least common denominator, so that the numerators can be combined into a single expression.

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