Solving Rational Inequality (-x+5)/(x-7) A Step-by-Step Guide

Solving rational inequalities involves finding the values of the variable that make the inequality true. This process typically involves finding critical values, creating a sign chart, and determining the intervals where the inequality holds. In this article, we will delve into a step-by-step approach to solve the rational inequality $\frac{-x+5}{x-7} \geq 0$, graph the solution set on a real number line, and express the solution set in interval notation. Mastering these techniques is crucial for success in algebra and calculus, providing a solid foundation for tackling more advanced mathematical problems.

Understanding Rational Inequalities

Before diving into the solution, let's define what a rational inequality is. A rational inequality is an inequality that contains one or more rational expressions. A rational expression is a fraction where the numerator and denominator are polynomials. When we encounter such inequalities, our primary goal is to identify the range of values for the variable that satisfy the given condition. This involves a methodical approach to ensure we capture all possible solutions while avoiding common pitfalls, such as incorrectly handling the sign changes around critical points. The process may seem intricate at first, but with a clear understanding of the steps, you will be able to tackle these problems with confidence.

Steps to Solve Rational Inequalities

To effectively solve rational inequalities, follow these steps:

1. Find the critical values: These are the values that make the numerator or the denominator equal to zero. Critical values are essential because they divide the number line into intervals where the expression's sign remains constant. They represent potential points where the inequality changes from true to false or vice versa. Accurately identifying these values is the first crucial step in solving the inequality.
2. Create a sign chart: A sign chart is a visual tool that helps determine the sign of the rational expression in each interval created by the critical values. This chart typically includes the critical values marked on a number line, with intervals between them. By testing a value within each interval, we can determine whether the expression is positive or negative in that interval. The sign chart provides a clear and organized way to analyze the behavior of the rational expression across different intervals.
3. Determine the intervals where the inequality holds: Based on the sign chart and the inequality, identify the intervals where the inequality is true. Consider whether the critical values themselves should be included in the solution set, depending on the inequality symbol ($\geq$, $\leq$, >, <). For inequalities involving $\geq$ or $\leq$, we include the critical values that make the expression equal to zero (unless they also make the denominator zero). For inequalities with > or <, we exclude the critical values. This step requires careful attention to detail to ensure the correct solution set is identified.
4. Express the solution set in interval notation: This is a concise way to represent the solution, using intervals and brackets to indicate the range of values that satisfy the inequality. Interval notation uses parentheses for open intervals (excluding endpoints) and square brackets for closed intervals (including endpoints). For example, (a, b) represents all values between a and b, excluding a and b, while [a, b] represents all values between a and b, including a and b. Mastering interval notation allows for clear and precise communication of the solution set.

Solving the Inequality $\frac{-x+5}{x-7} \geq 0$

Now, let’s apply these steps to solve the given rational inequality:

$\frac{-x+5}{x-7} \geq 0$

Step 1: Find the Critical Values

The critical values are found by setting the numerator and the denominator equal to zero.

Numerator:

$-x + 5 = 0$

$-x = -5$

$x = 5$

Denominator:

$x - 7 = 0$

$x = 7$

Thus, the critical values are $x = 5$ and $x = 7$. These values are crucial as they delineate the intervals where the sign of the expression may change. The value $x = 5$ makes the numerator zero, while $x = 7$ makes the denominator zero. Because division by zero is undefined, $x = 7$ will be a critical boundary that is not included in the solution set, even if the inequality symbol is $\geq$ or $\leq$. By identifying these critical values, we set the stage for a detailed examination of the inequality's behavior.

Step 2: Create a Sign Chart

Construct a number line and mark the critical values $5$ and $7$ on it. These points divide the number line into three intervals: $(-\infty, 5)$, $(5, 7)$, and $(7, \infty)$. To determine the sign of the expression in each interval, we will choose a test value from within each interval and substitute it into the rational expression.

• Interval $(-\infty, 5)$: Choose $x = 0$

  $\frac{-0+5}{0-7} = \frac{5}{-7} < 0$

• Interval $(5, 7)$: Choose $x = 6$

  $\frac{-6+5}{6-7} = \frac{-1}{-1} = 1 > 0$

• Interval $(7, \infty)$: Choose $x = 8$

  $\frac{-8+5}{8-7} = \frac{-3}{1} < 0$

Now, we can create the sign chart:

This sign chart clearly illustrates the sign of the expression $\frac{-x+5}{x-7}$ in each interval. In the interval $(-\infty, 5)$, the expression is negative. In the interval $(5, 7)$, the expression is positive. And in the interval $(7, \infty)$, the expression is again negative. This structured approach allows us to visually track how the expression changes across different intervals.

Step 3: Determine the Intervals Where the Inequality Holds

We are looking for the intervals where $\frac{-x+5}{x-7} \geq 0$. From the sign chart, we see that the expression is positive in the interval $(5, 7)$. Additionally, the expression equals zero when $x = 5$. Since the inequality includes “equal to,” we include $x = 5$ in the solution. However, $x = 7$ is not included because it makes the denominator zero, which is undefined.

Thus, the inequality holds in the interval $[5, 7)$. This careful consideration of both the positive intervals and the critical points is essential for determining the precise solution set. Including or excluding endpoints based on the inequality symbol and the behavior of the rational expression is a key aspect of solving these problems.

Step 4: Express the Solution Set in Interval Notation

The solution set in interval notation is $[5, 7)$. This notation clearly communicates that the solution includes all values from 5 up to, but not including, 7. The square bracket “[“ indicates that 5 is included in the solution, while the parenthesis “)” indicates that 7 is not included. The interval notation provides a concise and standardized way to represent the range of values that satisfy the inequality, making it easy to communicate the solution to others.

Graphing the Solution Set on a Real Number Line

To graph the solution set on a real number line, we mark the critical values and shade the intervals where the inequality holds. For the solution set $[5, 7)$:

• Place a closed circle (or a bracket) at $x = 5$ to indicate that $5$ is included in the solution.
• Place an open circle (or a parenthesis) at $x = 7$ to indicate that $7$ is not included in the solution.
• Shade the region between $5$ and $7$.

This visual representation provides an intuitive understanding of the solution set, clearly showing the range of values that satisfy the inequality. The closed circle at 5 and the open circle at 7 visually communicate the inclusion and exclusion of these endpoints, respectively. The shaded region between these points represents all the values that make the rational inequality true. Graphing the solution set is a helpful way to confirm your algebraic solution and ensure a comprehensive understanding of the problem.

Conclusion

In this article, we successfully solved the rational inequality $\frac{-x+5}{x-7} \geq 0$ by following a structured approach. We identified the critical values, created a sign chart, determined the intervals where the inequality holds, and expressed the solution set in interval notation as $[5, 7)$. We also graphed the solution set on a real number line, providing a visual representation of the solution. Mastering the techniques discussed here is crucial for anyone studying algebra, calculus, or related fields. By understanding the steps involved in solving rational inequalities, you can approach these problems with confidence and accuracy. Remember to practice these techniques with different examples to solidify your understanding and enhance your problem-solving skills.