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. Step 1: Understand the given rational function: \(\frac{2(X-2)}{(X-1)(X-3)} = 0\). To solve this equation, we need to find the values of \(X\) that make the numerator zero, since a fraction equals zero only when its numerator is zero (and the denominator is not zero).
2. Step 2: Set the numerator equal to zero: \$2(X-2) = 0\(. Since 2 is a constant and not zero, focus on \)X-2 = 0$.
3. Step 3: Solve for \(X\): \(X - 2 = 0 \implies X = 2\). This is the candidate solution where the function could be zero.
4. Step 4: Check the denominator at \(X=2\) to ensure it is not zero: \((2-1)(2-3) = (1)(-1) = -1 \neq 0\). Since the denominator is not zero, \(X=2\) is a valid solution.
5. Step 5: Use the graph to confirm the solution: Look at the graph at \(X=2\) and verify that the function crosses the x-axis (i.e., \(y=0\)) at this point. This confirms the solution found algebraically.

Key Concepts

Rational Functions and Their Graphs

A rational function is a ratio of two polynomials, and its graph can have vertical asymptotes where the denominator is zero. Understanding how to interpret these graphs helps identify values where the function is undefined and analyze behavior near these points.

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. This identifies the x-values where the function equals zero, while excluding points that make the denominator zero to avoid undefined values.

Interval Notation and Inequalities

Interval notation expresses solution sets for inequalities clearly, showing ranges of x-values where the function satisfies conditions like being greater or less than zero. Using the graph, one can determine these intervals by observing where the function lies above or below the x-axis.