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 occur where the numerator or denominator is zero: \(X=2\) (numerator zero), \(X=1\) and \(X=3\) (denominator zero).
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 where the function is above the x-axis (positive) or 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 denominator is zero, and 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 the expression to zero or another value. Solving them requires finding where the rational expression is positive, negative, or zero, often by analyzing critical points where the numerator or denominator is zero.

Critical Points and Domain Restrictions

Critical points occur where the numerator or denominator of a rational expression equals zero. These points divide the number line into intervals to test the inequality. Domain restrictions arise because the denominator cannot be zero, so these points are excluded from the solution set.

Graph Interpretation for Inequalities

Graphs of rational functions help visualize where the function is positive or negative. By examining the graph, one can identify intervals where the curve lies above or below the x-axis, corresponding to positive or negative values, which aids in solving inequalities.