Problem 33 - College Algebra

Question

Solve each rational inequality in Exercises \(29-48,\) 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

Based upon the analysis, the endpoints x = 2 and x = 4 are included in the solution. The intervals that satisfy the inequality are (2, 4) and (4, ∞), with the endpoint at x = 2 excluded, hence in interval notation: (2, 4] U [4, ∞)

Step 1: Identify Critical Points

Set the rational expression \(\frac{-x+2}{x-4}\) equal to zero and solve for 'x' by using cross multiplication. The bottom of the fraction, \(x-4\), cannot be zero because this would make the expression undefined which is not allowed in mathematics. Therefore, the critical points are the values of 'x' where the function equals to zero or is undefined.

Step 2: Solve for x Where y Equals Zero

Solving for \(x\) when the inequality is equal to zero gives: -x + 2 = 0. Solving this gives x = 2. However, when the denominator equals zero, x will be 4. Hence, the critical points are x = 2, x = 4.

Step 3: Analyze the Sign in Each Interval

Divide the number line into intervals using the critical points. This will give 3 intervals: (-∞, 2), (2, 4) and (4, ∞). Substitute a convenient value from each interval into the inequality to determine whether that interval satisfies the inequality or not. The intervals that satisfy the inequality will be included in the final answer.

Step 4: Write the Solution in Interval Notation

The solution in interval notation will be based on the intervals that satisfy the inequality from Step 3. Since this inequality includes the equal to sign, make sure the endpoints of the intervals are included (closed brackets) if the original inequality was satisfied at those points.

Key Concepts

Critical Points

Critical points in a rational inequality are the values of 'x' where the rational expression either equals zero or becomes undefined. These points are crucial because they divide the number line into different intervals that need to be tested separately.
To find critical points, follow these steps:

• First, set the numerator of the rational expression equal to zero and solve for 'x'. This helps find where the inequality can be zero. In our example, solving \[-x + 2 = 0\] gives us the critical point \(x = 2\).
• Next, identify where the denominator can become zero, making the expression undefined. For this expression, the denominator \[x - 4 = 0\] indicates another critical point at \(x = 4\).

These critical points play a pivotal role in dividing the number line for further testing.

Interval Notation

Interval notation is a concise way of representing intervals on the number line where a given condition is met. It is essential for expressing solution sets for inequalities.
The notation uses brackets to signify whether endpoints are included or not:

• Closed Bracket \([ \, ]\): Indicates that an endpoint is included in the interval (e.g., \([2, 4]\)).
• Open Bracket \(( \, )\): Indicates that an endpoint is not included (e.g., \((2, 4)\)).

In our inequality, since it includes \( \geq \), which means 'greater than or equal to,' closed brackets will be used for points that do not make the rational expression undefined.

Number Line Graph

Plotting on a number line graph visually represents the solution intervals discovered through testing critical points. This step helps to clearly see which parts of the number line satisfy the inequality.
To draw a number line graph:

• Mark the critical points on the line. In this exercise, they are \(x = 2\) and \(x = 4\).
• Test the intervals created by these points. For this inequality, the number line is divided into \((-\infty, 2),\ (2, 4), \ (4, \infty)\).
• Shade or indicate the intervals that satisfy the inequality. Based on testing, we'll highlight the valid intervals on the line.
  

This visual aid complements the written solution, making it easier to understand the mathematical relationship.

Solution Set

The solution set of an inequality consists of all the values of 'x' that satisfy the inequality condition. This set is usually expressed in interval notation.
For the rational inequality \[\frac{-x+2}{x-4} \geq 0\], testing the intervals between the critical points \((- \infty, 2), (2, 4), (4, \infty)\) reveals which intervals satisfy the condition. By checking a test point from each interval, we establish whether the sign of the expression is non-negative at those points.
Finally, intervals that meet the condition are included in the solution set. Care has to be taken with endpoints, checking whether the inequality allows them to be included (using closed brackets) or not (using open brackets). In this case, \(x = 2\) is included while \(x = 4\) is not, because it would make the denominator zero.
The solution set, written in interval notation, effectively communicates the complete range of satisfying values for the inequality.