Problem 47 - 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. $$ \frac{-x+2}{x-4} \geq 0 $$

Step-by-Step Solution

Answer

The solution is \(x \in (2, 4)\). It is represented by an open interval on the number line between 2 and 4.

Step 1: Identify the Critical Points

The inequality will change its sign either at the roots of the numerator or at the points where the denominator is equal to zero. To find these points, separate the rational into two parts: try \(x = 2\) from \( -x + 2 = 0\) where the function is either equal to zero, and \(x = 4\) from \(x - 4 = 0\) where the function is undefined.

Step 2: Testing the Intervals

Now, divide the number line into three regions based on the critical points: (-∞, 2), (2, 4), (4, ∞). Choose a test point in each interval and substitute it into the inequality. For (-∞, 2), choose \(x = 0\), this gives \(\frac{1}{-4}\) which is less than 0 hence this interval does not satisfy the inequality. For (2, 4) choose \(x = 3\) and it's easy to see that \(\frac{-1}{-1}\) is greater than 0, thus this interval is part of the solution set. For (4, ∞), choose \(x = 5\), we get \(\frac{-3}{1}\) which is less than 0 so this interval is also not part of the solution.

Step 3: Writing the solution in Interval Notation

The only interval that satisfies the inequality is (2, 4). In interval notation, this can be written as (2, 4).

Step 4: Graphing the Solution

On a number line, make a circle at \(x = 2\) and \(x = 4\), these will be open circles because 2 and 4 are not included in the solution set. Then draw a solid line between \(x = 2\) and \(x = 4\) to indicate that every point within this interval is a part of the solution.