Solve Inequality 5/(x-2) >= X-1 Step-by-Step Solution

In the realm of mathematics, inequalities play a crucial role in defining relationships between quantities that are not necessarily equal. Solving inequalities, especially those involving rational expressions, requires a meticulous approach to ensure accurate results. This article delves into the step-by-step process of solving the inequality ${\frac{5}{x-2} \ge x-1}$, providing a comprehensive guide for students and enthusiasts alike. Let's embark on this mathematical journey, breaking down each step with clarity and precision.

Understanding the Inequality

The given inequality, ${\frac{5}{x-2} \ge x-1}$, is a rational inequality because it involves a rational expression (a fraction with a variable in the denominator). To solve it, we need to find all values of ${x}$ that satisfy the inequality. This process involves several steps, including rearranging the inequality, finding critical points, and testing intervals.

Step 1: Rearrange the Inequality

The first step in solving this inequality is to rearrange it so that one side is zero. This makes it easier to analyze the sign of the expression. We achieve this by subtracting ${(x-1)}$ from both sides of the inequality:

${\frac{5}{x-2} - (x-1) \ge 0}$

Step 2: Combine into a Single Fraction

Next, we need to combine the terms on the left side into a single fraction. To do this, we find a common denominator, which in this case is ${(x-2)}$. We rewrite ${(x-1)}$ as a fraction with this denominator:

${\frac{5}{x-2} - \frac{(x-1)(x-2)}{x-2} \ge 0}$

Now, we can combine the fractions:

${\frac{5 - (x-1)(x-2)}{x-2} \ge 0}$

Step 3: Simplify the Numerator

We simplify the numerator by expanding the product and combining like terms:

${\frac{5 - (x^2 - 3x + 2)}{x-2} \ge 0}$

${\frac{5 - x^2 + 3x - 2}{x-2} \ge 0}$

${\frac{-x^2 + 3x + 3}{x-2} \ge 0}$

Step 4: Multiply by -1 (and Reverse the Inequality Sign)

To make the leading coefficient positive, we multiply both sides of the inequality by -1. Remember that multiplying or dividing an inequality by a negative number reverses the inequality sign:

${\frac{x^2 - 3x - 3}{x-2} \le 0}$

Step 5: Find the Critical Points

Critical points are the values of ${x}$ where the expression on the left side is either equal to zero or undefined. These points divide the number line into intervals that we will test later.

Finding Zeros of the Numerator

The zeros of the numerator are the solutions to the quadratic equation ${x^2 - 3x - 3 = 0}$. We can find these using the quadratic formula:

${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$

In this case, ${a = 1}$, ${b = -3}$, and ${c = -3}$. Plugging these values into the formula, we get:

${x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-3)}}{2(1)}}$

${x = \frac{3 \pm \sqrt{9 + 12}}{2}}$

${x = \frac{3 \pm \sqrt{21}}{2}}$

So, the zeros of the numerator are ${x_1 = \frac{3 - \sqrt{21}}{2}}$ and ${x_2 = \frac{3 + \sqrt{21}}{2}}$.

Finding Zeros of the Denominator

The denominator is ${x-2}$, so it is equal to zero when ${x = 2}$. This is another critical point.

Step 6: Create a Sign Chart

Now we create a sign chart to analyze the intervals determined by the critical points. Our critical points are ${x_1 = \frac{3 - \sqrt{21}}{2} \approx -0.79}$, ${x_2 = \frac{3 + \sqrt{21}}{2} \approx 3.79}$, and ${x = 2}$. These points divide the number line into four intervals:

1. ${(-\infty, \frac{3 - \sqrt{21}}{2})}$
2. ${(\frac{3 - \sqrt{21}}{2}, 2)}$
3. ${(2, \frac{3 + \sqrt{21}}{2})}$
4. ${(\frac{3 + \sqrt{21}}{2}, \infty)}$

We will test a value from each interval in the inequality ${\frac{x^2 - 3x - 3}{x-2} \le 0}$ to determine the sign of the expression in that interval.

Interval	Test Value (x)	x² - 3x - 3	x - 2	(x² - 3x - 3) / (x - 2)	≤ 0?
(–∞, (3 – √21) / 2)	–1	1 + 3 – 3 = 1	–3	–1 / 3	Yes
((3 – √21) / 2, 2)	0	–3	–2	3 / 2	No
(2, (3 + √21) / 2)	3	9 – 9 – 3 = –3	1	–3	Yes
((3 + √21) / 2, ∞)	4	16 – 12 – 3 = 1	2	1 / 2	No

Step 7: Determine the Solution Set

From the sign chart, we see that the inequality ${\frac{x^2 - 3x - 3}{x-2} \le 0}$ is satisfied in the intervals ${(-\infty, \frac{3 - \sqrt{21}}{2}]}$ and ${(2, \frac{3 + \sqrt{21}}{2}]}$. Note that we include the zeros of the numerator because the inequality is non-strict (${\le}$), but we exclude ${x = 2}$ because it makes the denominator zero.

Therefore, the solution set is:

${x \in \left(-\infty, \frac{3 - \sqrt{21}}{2}\right] \cup \left(2, \frac{3 + \sqrt{21}}{2}\right]}$

Expressing the Solution

The solution can be expressed in interval notation as:

${\left(-\infty, \frac{3 - \sqrt{21}}{2}\right] \cup \left(2, \frac{3 + \sqrt{21}}{2}\right]}$

This means that the inequality holds for all values of ${x}$ less than or equal to ${\frac{3 - \sqrt{21}}{2}}$ or for values of ${x}$ greater than 2 but less than or equal to ${\frac{3 + \sqrt{21}}{2}}$.

Graphical Interpretation

A graphical interpretation can further clarify the solution. The graph of the function ${f(x) = \frac{x^2 - 3x - 3}{x-2}}$ will be less than or equal to zero in the intervals we found. The critical points are where the graph crosses the x-axis (zeros of the numerator) and where there is a vertical asymptote (zero of the denominator).

Conclusion

Solving the inequality ${\frac{5}{x-2} \ge x-1}$ involves a series of algebraic manipulations and careful analysis. By rearranging the inequality, finding critical points, creating a sign chart, and testing intervals, we can determine the solution set. The solution is:

${x \in \left(-\infty, \frac{3 - \sqrt{21}}{2}\right] \cup \left(2, \frac{3 + \sqrt{21}}{2}\right]}$

This comprehensive guide provides a clear understanding of the steps involved in solving rational inequalities, a fundamental concept in mathematics. Understanding how to solve inequalities is crucial for various applications in calculus, algebra, and other mathematical fields. By mastering these techniques, students can tackle a wide range of problems involving inequalities and deepen their understanding of mathematical concepts.

The given problem is to solve for x in the inequality ${\frac{5}{x-2} \ge x-1}$. This inequality involves a rational expression, making the solution process somewhat intricate. The main goal is to find all values of ${x}$ that satisfy the inequality. Solving inequality problems such as this often requires a combination of algebraic manipulation and careful consideration of interval testing. Before diving into the detailed steps, it’s essential to understand the key concepts involved in solving inequalities, particularly those that involve rational functions. This approach ensures that no values are missed and that the solutions are accurate.

Initial Rearrangement and Simplification

The first step in solving the inequality ${\frac{5}{x-2} \ge x-1}$ is to rearrange it such that one side of the inequality is zero. This can be done by subtracting ${(x-1)}$ from both sides, leading to ${\frac{5}{x-2} - (x-1) \ge 0}$. This form is easier to work with as it allows us to combine terms and analyze the sign of the expression more effectively. Next, the terms on the left side need to be combined into a single fraction. To achieve this, we express ${(x-1)}$ with a common denominator, which is ${(x-2)}$. The expression becomes ${\frac{5}{x-2} - \frac{(x-1)(x-2)}{x-2} \ge 0}$. Combining the fractions, we get ${\frac{5 - (x-1)(x-2)}{x-2} \ge 0}$. This process streamlines the inequality, making it simpler to identify critical points later on. Proper algebraic manipulation at this stage is crucial for solving complex inequalities and achieving the correct solution set.

Expanding and Simplifying the Numerator

After combining the fractions, the next crucial step is to simplify the numerator. This involves expanding the product ${(x-1)(x-2)}$ and then combining like terms. Expanding ${(x-1)(x-2)}$ gives ${x^2 - 3x + 2}$. Substituting this back into the numerator, the inequality becomes ${\frac{5 - (x^2 - 3x + 2)}{x-2} \ge 0}$. Distributing the negative sign, we have ${\frac{5 - x^2 + 3x - 2}{x-2} \ge 0}$. Combining like terms in the numerator simplifies the expression to ${\frac{-x^2 + 3x + 3}{x-2} \ge 0}$. To make the leading coefficient positive, we multiply both sides of the inequality by -1. Remember that this step also reverses the inequality sign, resulting in ${\frac{x^2 - 3x - 3}{x-2} \le 0}$. This transformation is a common practice when solving inequalities to make subsequent steps, such as finding critical points, more straightforward. The resulting quadratic expression in the numerator is now in a standard form, making it easier to apply techniques like the quadratic formula.

Finding Critical Points: Zeros of Numerator and Denominator

To effectively solve the inequality, identifying critical points is essential. Critical points are the values of ${x}$ where the expression on the left side of the inequality is either equal to zero or undefined. These points serve as boundaries that divide the number line into intervals, which we will later test to determine the solution set. The critical points come from two sources: the zeros of the numerator and the zeros of the denominator. The zeros of the numerator are the solutions to the equation ${x^2 - 3x - 3 = 0}$. Since this is a quadratic equation, we can use the quadratic formula to find the solutions. The quadratic formula is given by ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$, where ${a}$, ${b}$, and ${c}$ are the coefficients of the quadratic equation. In this case, ${a = 1}$, ${b = -3}$, and ${c = -3}$. Plugging these values into the formula, we get ${x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-3)}}{2(1)}}$. Simplifying, we find ${x = \frac{3 \pm \sqrt{21}}{2}}$. Therefore, the zeros of the numerator are ${x_1 = \frac{3 - \sqrt{21}}{2}}$ and ${x_2 = \frac{3 + \sqrt{21}}{2}}$. The zeros of the denominator occur where the denominator, ${x-2}$, is equal to zero. This happens when ${x = 2}$. Thus, the critical points are ${x_1 = \frac{3 - \sqrt{21}}{2}}$, ${x_2 = \frac{3 + \sqrt{21}}{2}}$, and ${x = 2}$. These critical points are pivotal in setting up the intervals for further analysis.

Constructing and Analyzing the Sign Chart

After identifying the critical points, the next step in solving the inequality is to construct a sign chart. The sign chart helps to determine the intervals where the inequality ${\frac{x^2 - 3x - 3}{x-2} \le 0}$ holds true. The critical points, ${\frac{3 - \sqrt{21}}{2}}$, ${2}$, and ${\frac{3 + \sqrt{21}}{2}}$, divide the number line into four intervals:

1. ${(-\infty, \frac{3 - \sqrt{21}}{2})}$
2. ${(\frac{3 - \sqrt{21}}{2}, 2)}$
3. ${(2, \frac{3 + \sqrt{21}}{2})}$
4. ${(\frac{3 + \sqrt{21}}{2}, \infty)}$

For each interval, we choose a test value and substitute it into the inequality to determine the sign of the expression. Let’s analyze each interval:

Interval 1: ${(-\infty, \frac{3 - \sqrt{21}}{2})}$

Choose a test value, such as ${x = -1}$. Substituting ${x = -1}$ into the inequality, we get:

${\frac{(-1)^2 - 3(-1) - 3}{-1 - 2} = \frac{1 + 3 - 3}{-3} = \frac{1}{-3}}$

The result is negative, so the inequality holds true in this interval.

Interval 2: ${(\frac{3 - \sqrt{21}}{2}, 2)}$

Choose a test value, such as ${x = 0}$. Substituting ${x = 0}$ into the inequality, we get:

${\frac{(0)^2 - 3(0) - 3}{0 - 2} = \frac{-3}{-2} = \frac{3}{2}}$

The result is positive, so the inequality does not hold true in this interval.

Interval 3: ${(2, \frac{3 + \sqrt{21}}{2})}$

Choose a test value, such as ${x = 3}$. Substituting ${x = 3}$ into the inequality, we get:

${\frac{(3)^2 - 3(3) - 3}{3 - 2} = \frac{9 - 9 - 3}{1} = -3}$

The result is negative, so the inequality holds true in this interval.

Interval 4: ${(\frac{3 + \sqrt{21}}{2}, \infty)}$

Choose a test value, such as ${x = 4}$. Substituting ${x = 4}$ into the inequality, we get:

${\frac{(4)^2 - 3(4) - 3}{4 - 2} = \frac{16 - 12 - 3}{2} = \frac{1}{2}}$

The result is positive, so the inequality does not hold true in this interval.

Identifying the Solution Set

Based on the sign chart analysis, the inequality ${\frac{x^2 - 3x - 3}{x-2} \le 0}$ is satisfied in the intervals ${(-\infty, \frac{3 - \sqrt{21}}{2}]}$ and ${(2, \frac{3 + \sqrt{21}}{2}]}$. It’s important to note that we include the zeros of the numerator (${\frac{3 - \sqrt{21}}{2}}$ and ${\frac{3 + \sqrt{21}}{2}}$) because the inequality is non-strict (${\le}$). However, we exclude ${x = 2}$ because it makes the denominator zero, resulting in an undefined expression. Therefore, the solution set is:

${x \in \left(-\infty, \frac{3 - \sqrt{21}}{2}\right] \cup \left(2, \frac{3 + \sqrt{21}}{2}\right]}$

This is the complete solution to the inequality, expressed in interval notation. It signifies all the values of ${x}$ for which the original inequality holds true. Solving such inequalities reinforces understanding of algebraic manipulations and interval analysis, vital skills in mathematics. Understanding inequality solution sets is crucial for various mathematical applications and problem-solving scenarios.