Use the graph to solve each equation or inequality. Use interval notation where appropriate. 2(x-2) / {(x-1)(x-3)} = 0

Verified step by step guidance

1. Identify the given rational function: \(\frac{2(X-2)}{(X-1)(X-3)} = 0\).
2. Recall that a rational function equals zero when its numerator equals zero (and the denominator is not zero). So, set the numerator equal to zero: \$2(X-2) = 0$.
3. Solve the numerator equation: \$2(X-2) = 0\( implies \)X-2 = 0\(, so \)X = 2$.
4. Check the denominator at \(X=2\) to ensure it is not zero: \((2-1)(2-3) = 1 \times (-1) = -1 \neq 0\), so \(X=2\) is a valid solution.
5. Use the graph to confirm that the function crosses the x-axis at \(X=2\), which matches the solution found algebraically.

Key Concepts

Rational Functions and Their Graphs

A rational function is a ratio of two polynomials. Its graph can have vertical asymptotes where the denominator is zero and horizontal or oblique asymptotes based on the degrees of numerator and denominator. Understanding these features helps interpret the behavior of the function and solve related equations.

Solving Rational Equations

To solve rational equations like 2(x-2)/((x-1)(x-3)) = 0, set the numerator equal to zero and ensure the denominator is not zero. The solutions are the x-values that make the numerator zero but do not make the denominator zero, as those points are undefined.

Interval Notation and Inequalities

Interval notation expresses the set of solutions for inequalities or equations on the number line. When solving inequalities involving rational functions, consider where the function is positive, negative, or zero, and exclude points where the function is undefined, using parentheses or brackets accordingly.