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 \(\frac{2(X-2)}{(X-1)(X-3)}\) by setting the numerator and denominator equal to zero. The numerator \(2(X-2) = 0\) gives \(X=2\), and the denominator \((X-1)(X-3) = 0\) gives \(X=1\) and \(X=3\). These points divide the number line into intervals.
2. Determine the sign of the function on each interval created by the critical points \((-\infty, 1)\), \((1, 2)\), \((2, 3)\), and \((3, \infty)\). Use test points from each interval and substitute them into the function to check if the function is positive or negative there.
3. Analyze the graph to confirm the sign of the function on each interval. Notice where the graph is above the x-axis (function positive) and where it is below the x-axis (function negative).
4. Since the inequality is \(\frac{2(X-2)}{(X-1)(X-3)} \leq 0\), select the intervals where the function is less than or equal to zero. Include points where the function equals zero (roots of the numerator) but exclude points where the function is undefined (roots of the denominator).
5. Express the solution in interval notation by combining the intervals where the inequality holds true, making sure to exclude \(X=1\) and \(X=3\) where the function is undefined.

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, 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 function 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.

Using Graphs to Solve Inequalities

Graphs visually represent where a function is positive, negative, or zero. For rational inequalities, the graph helps identify intervals where the function lies below or on the x-axis (≤ 0). This visual approach simplifies understanding the solution set in interval notation.