## College Algebra (6th Edition)

$(1, 2)$
Consider the Rational Inequality as follows: $\frac{x}{x-1}>2$ Here are the steps required for Solving Rational Inequalities: Step 1: One side must be zero and the other side can have only one fraction, so we simplify the fractions if there is more than one fraction. Let us subtract 2 from both sides to obtain zero on the right. $\frac{x}{x-1}>2$ $\frac{(x)}{x-1}-2>0$ $\frac{(x-2(x-1)}{x-1}>0$ $\frac{-x+2}{x-1}>0$ Step 2: Critical or Key Values are first evaluated. In order to this, set the numerator and denominator of the fraction equal to zero and then simplified rational inequality is solved. $-x+2 = 0$ This implies $x =2$ And $x-1=0$ This implies $x =1$ These solutions are used as boundary points on a number line. Step 3: Locate the boundary points on a number line found in Step 2 to divide the number line into intervals. The boundary points divide the number line into three intervals: $(-∞,1), (1, 2), (2, ∞)$ Step 4: Now, one test value within each interval is chosen and $f$ is evaluated at that number. Intervals: $(-∞,1), (1, 2), (2, ∞)$ Test value: $0$ $1.5$ $3$ Sign Change: Negative Positive Negative $f (x)> 0?$: F T F Step 5: Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x)> 0$ . Based on our work done in Step 4, we see that $f (x)> 0$ for all x in $(1, 2)$ . Conclusion: Thus, the interval notation of the given inequality is $(1, 2)$ and the graph of the solution set on a number line is shown as follows: