Problem 58 - Algebra and Trigonometry

Question

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{1}{x-3}<1 $$

Step-by-Step Solution

Answer

The solution set in interval notation is (3, ∞)

Step 1: Rewrite the Inequality

A common first step in these types of problems is to make the inequality easier to solve. That can often mean getting the variable by itself on one side. To do this, subtract 1 from each side of the equation: \(\frac{1}{x-3}-1<0\). It can be further simplified to \(\frac{1-x +3}{x-3}<0\), which is \(\frac{-x + 4}{x - 3}<0\) after combining like terms.

Step 2: Find the Critical Points

The critical points of a rational function are the values of 'x' that make the numerator 0 and the values that make the denominator 0. We need these critical points to solve the inequality. To find the critical points, set the numerator and the denominator separately equal to 0 and solve for 'x'. So, for numerator -x+4=0 gives 'x = 4' as a critical point and for the denominator, x-3=0 gives 'x = 3' as a critical point.

Step 3: Test Intervals

Since we have two critical points, 3 and 4, they divide the number line into three intervals: (-∞, 3), (3, 4), and (4, ∞). Select a test point from each of these intervals and substitute into the inequality and check if the inequality holds. Choosing 'x = 2' for (-∞, 3), 'x = 3.5' for (3, 4), and 'x = 5' for (4, ∞) can fulfill the conditions. Substituting these values into the inequality, we find that the intervals (3, 4) and (4, ∞) satisfy the inequality.

Step 4: Write the Solution in Interval Notation

Now that the solution intervals have been found, write the solution set to the inequality in interval notation. Here, the solution set is (3, 4) and (4, ∞). The notation (a, b) represents all numbers between 'a' and 'b'. The notation (a, ∞) represents all numbers greater than 'a'. Therefore, the solution in interval notation is (3, ∞).