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 rational expression \(\frac{2(X-2)}{(X-1)(X-3)}\). These points are where the numerator or denominator equals zero: \(X=2\) (numerator zero), \(X=1\) and \(X=3\) (denominator zero, vertical asymptotes).
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 expression \(\frac{2(X-2)}{(X-1)(X-3)}\) on each interval by choosing a test point from each interval and substituting it into the expression.
4. Use the graph to verify the sign of the function on each interval. The graph shows the function's behavior around the vertical asymptotes at \(X=1\) and \(X=3\) and the zero at \(X=2\).
5. Select the intervals where the expression is less than zero (negative) and write the solution in interval notation, excluding points where the denominator is zero (since the function is undefined there).

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 Sign Analysis

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, and testing each interval helps determine where the inequality holds true.

Graph Interpretation of Rational Functions

Graphs of rational functions show vertical asymptotes at values making the denominator zero and zeros where the numerator is zero. By examining the graph, one can identify intervals where the function is above or below the x-axis, aiding in solving inequalities.