Solving (x-8)^2/(x^2-49) ≥ 0 Algebraically A Step-by-Step Guide

In the realm of mathematics, solving inequalities is a fundamental skill with applications spanning various fields. This article delves into the process of algebraically solving the inequality (x-8)^2/(x2-49) ≥ 0, providing a step-by-step guide and insightful explanations to enhance your understanding. We will explore the critical values, test intervals, and the significance of the inequality sign, equipping you with the tools to confidently tackle similar problems.

Understanding the Inequality: (x-8)^2/(x2-49) ≥ 0

At its core, the inequality (x-8)^2/(x2-49) ≥ 0 asks us to find the values of x for which the expression on the left-hand side is greater than or equal to zero. This means we are looking for the intervals where the expression is either positive or zero. To effectively solve this inequality, we need to consider the interplay between the numerator, (x-8)^2, and the denominator, x^2-49. The numerator, being a squared term, is always non-negative. However, the denominator, x^2-49, can be positive, negative, or zero, influencing the overall sign of the expression. A critical aspect of solving inequalities involving rational expressions is identifying the critical values. These are the values of x that make either the numerator or the denominator equal to zero. The critical values serve as boundaries that divide the number line into intervals, within which the expression maintains a consistent sign. By analyzing the sign of the expression within each interval, we can determine the solution set of the inequality. The numerator, (x-8)^2, becomes zero when x = 8, while the denominator, x^2-49, becomes zero when x = ±7. These three values, -7, 7, and 8, are our critical values. They are the cornerstones for constructing our test intervals and ultimately finding the solution to the inequality. It is crucial to remember that values that make the denominator zero are excluded from the solution set, as they would lead to an undefined expression. Therefore, x = -7 and x = 7 will not be included in the final answer. The square in the numerator (x-8)^2 indicates that the expression will be zero at x=8, and it will always be positive for any other value of x. This knowledge simplifies the analysis as we know the numerator's sign behavior. The denominator, x^2 - 49, can be factored into (x-7)(x+7), which allows us to easily identify the intervals where it is positive or negative. The sign of the denominator will directly impact the sign of the entire expression, given the numerator is non-negative. By understanding the behavior of both the numerator and the denominator, we can systematically determine the solution to the inequality.

Step-by-Step Solution

1. Identify Critical Values

The first crucial step in solving this inequality is to identify the critical values. These are the values of x that make either the numerator or the denominator equal to zero. The numerator, (x-8)^2, becomes zero when:x - 8 = 0 => x = 8The denominator, x^2 - 49, can be factored as (x - 7)(x + 7). Thus, it becomes zero when:x - 7 = 0 => x = 7 or x + 7 = 0 => x = -7Therefore, our critical values are -7, 7, and 8. These values are pivotal as they divide the number line into intervals where the expression's sign remains consistent. Understanding the nature of these critical values is essential for accurately determining the solution set. The critical values from the numerator (x=8) represent points where the expression equals zero, and depending on the inequality sign (≥ or >), these points may or may not be included in the solution. The critical values from the denominator (x=-7 and x=7) are particularly important because they represent values where the expression is undefined. These values must always be excluded from the solution set, as division by zero is undefined in mathematics. The identification of critical values is not just a mechanical step; it reflects a deeper understanding of the behavior of rational functions and how their signs change around these points. By finding these values, we are essentially mapping out the landscape of the inequality, which guides us toward the correct solution. Careful and accurate identification of critical values is thus the cornerstone of solving inequalities involving rational expressions.

2. Create Test Intervals

With the critical values identified (-7, 7, and 8), the next step is to create the test intervals. These intervals are formed by partitioning the number line using the critical values as boundaries. This process divides the number line into distinct regions, within each of which the expression (x-8)^2/(x2-49) maintains a consistent sign (either positive or negative). The intervals are:(-∞, -7), (-7, 7), (7, 8), and (8, ∞)Each interval represents a range of x-values that we will test to determine whether they satisfy the inequality. The endpoints of these intervals are the critical values themselves, which are crucial points where the expression can change its sign. It is important to note that the intervals are open intervals, meaning that the critical values themselves are not included within the interval. This is because we need to test these boundary points separately, as they may either make the expression equal to zero (in the case of critical values from the numerator) or undefined (in the case of critical values from the denominator). By creating these test intervals, we effectively break down the complex problem of solving the inequality over the entire number line into a series of smaller, more manageable tasks. We can then analyze the sign of the expression within each interval independently, which significantly simplifies the overall solution process. This systematic approach ensures that we do not overlook any potential solutions and that we accurately capture the complete solution set of the inequality.

3. Test Each Interval

Now, we select a test value within each interval and substitute it into the original inequality, (x-8)^2/(x2-49) ≥ 0, to determine the sign of the expression in that interval. This process will reveal whether the interval satisfies the inequality. * Interval (-∞, -7): Choose x = -8.(((-8)-8)^2)/((-8)2-49) = (256)/(15) > 0. The inequality holds.* Interval (-7, 7): Choose x = 0.((0-8)^2)/(02-49) = (64)/(-49) < 0. The inequality does not hold.* Interval (7, 8): Choose x = 7.5.((7.5-8)^2)/((7.5)2-49) = (0.25)/(7.25) > 0. The inequality holds.* Interval (8, ∞): Choose x = 9.((9-8)^2)/(92-49) = (1)/(32) > 0. The inequality holds.By meticulously testing each interval, we can accurately map the sign behavior of the expression across the number line. This is a critical step in solving inequalities because it allows us to identify the regions where the inequality is satisfied. The choice of test values within each interval is arbitrary, as the sign of the expression will remain consistent throughout the interval. However, it is generally advisable to choose simple values that are easy to compute, to minimize the risk of errors. The sign of the expression in each interval is determined by the combined effect of the numerator and the denominator. If both are positive or both are negative, the expression is positive. If one is positive and the other is negative, the expression is negative. The testing process not only tells us whether the inequality holds in a given interval but also provides valuable insight into the overall behavior of the expression. This understanding is crucial for accurately interpreting the results and for confidently stating the final solution set.

4. Determine the Solution Set

Based on the test results, we identify the intervals where the inequality (x-8)^2/(x2-49) ≥ 0 holds true. Recall that the inequality is satisfied when the expression is either positive or equal to zero. From our testing: * Interval (-∞, -7): The inequality holds.* Interval (-7, 7): The inequality does not hold.* Interval (7, 8): The inequality holds.* Interval (8, ∞): The inequality holds.Additionally, we need to consider the critical values themselves. * x = 8: The expression equals zero, so it satisfies the inequality. * x = -7 and x = 7: The expression is undefined, so they are not included in the solution.Therefore, the solution set includes the intervals (-∞, -7) and (7, 8), (8, ∞), as well as the point x = 8. Expressing this in interval notation, the solution is:(-∞, -7) ∪ (7, 8] ∪ [8, ∞)This final step is crucial as it synthesizes all the information gathered throughout the solution process into a concise and accurate representation of the solution set. The union symbol (∪) indicates that the solution set is the combination of all the individual intervals and points that satisfy the inequality. The use of parentheses and brackets is also important, as it indicates whether the endpoints are included or excluded from the solution set. In this case, the parentheses around -7 and 7 indicate that these values are not included, as they make the denominator zero and the expression undefined. The brackets around 8 indicate that this value is included, as it makes the expression equal to zero, which satisfies the “greater than or equal to” condition. The final solution set represents all the possible values of x that make the inequality true. It is a complete and precise answer that fully addresses the problem posed.

Final Answer

The solution to the inequality (x-8)^2/(x2-49) ≥ 0 in interval notation is:(-∞, -7) ∪ (7, ∞)

This solution represents the set of all real numbers x that satisfy the given inequality. The union of the intervals (-∞, -7) and (7, ∞) indicates that the inequality holds true for all values of x less than -7 or greater than 7. Note that the critical values -7 and 7 are excluded from the solution set, as they make the denominator of the expression equal to zero, resulting in an undefined value. The critical value 8, which makes the numerator zero, is included in the solution set because the inequality includes the