Question

Use the graph to solve each equation or inequality. Use interval notation where appropriate. 2(x-2) / {(x-1)(x-3)} ≤ 0

Accepted Answer

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. 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. 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). 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). 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.

Use the graph to solve each equation or inequality. Use interval notation where appropriate. 2(x-2) / {(x-1)(x-3)} ≤ 0

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Key Concepts

Rational Inequalities

Critical Points and Domain Restrictions

Using Graphs to Solve Inequalities