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 critical points of the function \( f(x) = \frac{2(x-2)}{(x-1)(x-3)} \). These occur where the numerator or denominator is zero: set \( 2(x-2) = 0 \) and \( (x-1)(x-3) = 0 \). This gives critical points at \( x = 2 \), \( x = 1 \), and \( x = 3 \).
2. Divide the number line into intervals based on these critical points: \( (-\infty, 1) \), \( (1, 2) \), \( (2, 3) \), and \( (3, \infty) \).
3. Determine the sign of the function \( f(x) \) on each interval by choosing a test point from each interval and substituting it into the function \( f(x) = \frac{2(x-2)}{(x-1)(x-3)} \).
4. Use the graph to verify the sign of the function on each interval. The graph shows where the function is above the x-axis (positive) and below the x-axis (negative).
5. Write the solution to the inequality \( \frac{2(x-2)}{(x-1)(x-3)} > 0 \) by including the intervals where the function is positive, excluding points where the function is undefined (vertical asymptotes at \( x=1 \) and \( x=3 \)). Express the solution in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the rational expression is positive, negative, or zero by analyzing the signs of numerator and denominator.

Critical Points and Domain Restrictions

Critical points occur where the numerator or denominator equals zero, causing the rational expression to be zero or undefined. These points divide the number line into intervals to test the inequality. Domain restrictions exclude values making the denominator zero, as the function is undefined there.

Graph Interpretation for Inequalities

Graphs of rational functions help visualize where the function is positive or negative by showing the curve's position relative to the x-axis. Intervals where the graph lies above the x-axis correspond to positive values, aiding in solving inequalities by identifying solution intervals.