Solving The Inequality 16/(x-3) < -4 A Step-by-Step Guide

This article provides a comprehensive guide to solving the inequality $\frac{16}{x-3}<-4$. Inequalities, a fundamental concept in mathematics, play a crucial role in various fields, including algebra, calculus, and real-world problem-solving. Mastering the techniques for solving inequalities is essential for students and professionals alike. This article will walk you through the process step by step, ensuring a clear understanding of the underlying principles and the correct solution.

Understanding Inequalities

Before diving into the specifics of the given inequality, let's first establish a solid understanding of what inequalities are and how they differ from equations. In mathematics, an inequality is a statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which state that two expressions are equal, inequalities indicate a range of possible values that satisfy the condition.

When solving inequalities, our goal is to isolate the variable on one side of the inequality symbol, similar to solving equations. However, there are some key differences in the techniques used. One crucial difference is that multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol. This is a critical rule to remember to ensure accurate solutions.

Another important consideration when dealing with inequalities, especially those involving rational expressions like the one we're addressing, is the identification of critical values. Critical values are the points where the expression is either undefined or equal to zero. These values divide the number line into intervals, and we need to test each interval to determine where the inequality holds true. These critical values are essential in identifying potential solutions and ensuring the final solution set is accurate.

Step 1: Rearrange the Inequality

Our initial inequality is $\frac{16}{x-3}<-4$. To begin, we need to rearrange the inequality so that one side is zero. This makes it easier to analyze the sign changes of the expression. We achieve this by adding 4 to both sides of the inequality:

$$\frac{16}{x-3} + 4 < 0$$

This step sets the stage for combining the terms on the left side and simplifying the inequality into a more manageable form. By having zero on one side, we can focus on the sign of the expression on the other side and determine the intervals where the inequality is satisfied.

Step 2: Combine Terms and Simplify

Now that we have $\frac{16}{x-3} + 4 < 0$, we need to combine the terms on the left side. To do this, we find a common denominator, which in this case is $(x-3)$. We rewrite 4 as $\frac{4(x-3)}{x-3}$ and add it to the fraction:

$$\frac{16}{x-3} + \frac{4(x-3)}{x-3} < 0$$

Next, we combine the numerators:

$$\frac{16 + 4(x-3)}{x-3} < 0$$

Now, we simplify the numerator:

$$\frac{16 + 4x - 12}{x-3} < 0$$

$$\frac{4x + 4}{x-3} < 0$$

Finally, we can factor out a 4 from the numerator:

$$\frac{4(x + 1)}{x-3} < 0$$

This simplified form makes it easier to identify the critical values and analyze the intervals where the inequality holds true. The simplified fraction allows us to clearly see the factors that affect the sign of the expression.

Step 3: Identify Critical Values

Critical values are the values of $x$ that make the expression either equal to zero or undefined. In our simplified inequality $\frac{4(x + 1)}{x-3} < 0$, the critical values occur when the numerator or the denominator is equal to zero.

First, let's find the values that make the numerator zero:

$$4(x + 1) = 0$$

$$x + 1 = 0$$

$$x = -1$$

Next, let's find the values that make the denominator zero:

$$x - 3 = 0$$

$$x = 3$$

So, our critical values are $x = -1$ and $x = 3$. These values divide the number line into three intervals: $(-\infty, -1)$, $(-1, 3)$, and $(3, \infty)$. We will test each interval to determine where the inequality $\frac{4(x + 1)}{x-3} < 0$ holds true. Identifying these critical values is a crucial step in solving inequalities, as they mark the points where the expression can change its sign.

Step 4: Test Intervals

To determine the solution set, we need to test a value from each interval in the inequality $\frac{4(x + 1)}{x-3} < 0$. The intervals are $(-\infty, -1)$, $(-1, 3)$, and $(3, \infty)$.

1. Interval $(-\infty, -1)$: Choose a test value, say $x = -2$. Substitute it into the simplified inequality:

   $$\frac{4(-2 + 1)}{-2 - 3} = \frac{4(-1)}{-5} = \frac{-4}{-5} = \frac{4}{5} > 0$$

   Since $\frac{4}{5}$ is not less than 0, this interval is not part of the solution.

2. Interval $(-1, 3)$: Choose a test value, say $x = 0$. Substitute it into the simplified inequality:

   $$\frac{4(0 + 1)}{0 - 3} = \frac{4(1)}{-3} = -\frac{4}{3} < 0$$

   Since $-\frac{4}{3}$ is less than 0, this interval is part of the solution.

3. Interval $(3, \infty)$: Choose a test value, say $x = 4$. Substitute it into the simplified inequality:

   $$\frac{4(4 + 1)}{4 - 3} = \frac{4(5)}{1} = 20 > 0$$

   Since 20 is not less than 0, this interval is not part of the solution.

By testing values in each interval, we can determine the intervals where the inequality holds true. This method ensures that we account for all possible solutions and avoid errors in our final answer. Testing each interval is a systematic approach to solving inequalities and provides a clear understanding of the solution set.

Step 5: Write the Solution

From our interval testing, we found that the inequality $\frac{4(x + 1)}{x-3} < 0$ is true for the interval $(-1, 3)$. Since the inequality is strictly less than (not less than or equal to), we use open intervals, which means we do not include the critical values $x = -1$ and $x = 3$ in the solution.

Therefore, the solution to the inequality $\frac{16}{x-3} < -4$ is:

$$-1 < x < 3$$

This can also be written in interval notation as $(-1, 3)$. This is the final solution to the inequality, representing the range of $x$ values that satisfy the given condition. Expressing the solution in both inequality and interval notation provides a clear and concise representation of the answer.

Conclusion

In this article, we have provided a detailed, step-by-step solution to the inequality $\frac{16}{x-3} < -4$. By rearranging the inequality, combining terms, identifying critical values, testing intervals, and writing the solution, we have demonstrated a comprehensive approach to solving rational inequalities. Remember, understanding the underlying concepts and applying the correct techniques are key to mastering inequalities. With practice, you can confidently solve a wide range of inequality problems. Inequalities are a crucial aspect of mathematical analysis, and proficiency in solving them opens doors to more advanced topics in mathematics and its applications.

Final Answer

The final answer is **B. $-1

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