Solving Exponential Inequalities A Step-by-Step Guide

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Introduction to Inequality Problems

In the realm of mathematics, inequality problems often present a fascinating challenge, requiring a blend of algebraic manipulation, logarithmic properties, and careful consideration of domain restrictions. This article delves into solving a specific inequality: 3 * 16((x2 - 29) / (-3x)) - 10 * 16((x2 - 29) / (-6x)) > 8. This problem exemplifies the intricate nature of exponential inequalities, where a keen understanding of exponential functions and their behavior is paramount. Our approach will involve transforming the inequality into a more manageable form, utilizing substitution techniques, and ultimately determining the solution set. Before diving into the specifics, it's crucial to appreciate the broader context of inequality problems in mathematics. Inequalities are not merely abstract exercises; they have practical applications in various fields, including optimization problems, economics, and computer science. Understanding how to solve inequalities is a fundamental skill that empowers us to model and analyze real-world scenarios where constraints and limitations play a significant role.

The given inequality, 3 * 16((x2 - 29) / (-3x)) - 10 * 16((x2 - 29) / (-6x)) > 8, at first glance, might seem daunting due to its exponential terms and the complex fraction in the exponent. However, by systematically applying algebraic techniques and leveraging the properties of exponents, we can unravel its intricacies. The key to solving this inequality lies in recognizing the common exponential term and employing a substitution to simplify the expression. This transformation will allow us to convert the inequality into a more familiar form, such as a quadratic inequality, which we can then solve using standard methods. Throughout the solution process, we must remain vigilant about potential domain restrictions, particularly those arising from the denominator in the exponent. These restrictions will play a crucial role in determining the final solution set. As we navigate through the steps, we will emphasize the underlying mathematical principles and provide clear explanations to enhance understanding. This will not only help in solving this particular inequality but also equip you with the tools to tackle similar problems in the future. The journey of solving this inequality is a testament to the power of mathematical reasoning and the elegance of problem-solving strategies. By carefully dissecting the problem and applying the appropriate techniques, we can arrive at a solution that not only satisfies the inequality but also deepens our appreciation for the interconnectedness of mathematical concepts. This process is akin to piecing together a puzzle, where each step builds upon the previous one, ultimately revealing the complete picture. So, let's embark on this mathematical adventure and uncover the solution to this intriguing inequality.

Step-by-Step Solution

1. Simplify the Exponential Terms

Our initial focus should be on simplifying the exponential terms within the inequality. We observe that the base of the exponent is 16, which can be expressed as 2^4. This substitution will be pivotal in streamlining the expression and revealing underlying relationships. Let's rewrite the inequality, paying close attention to the exponents:

3 * 16((x2 - 29) / (-3x)) - 10 * 16((x2 - 29) / (-6x)) > 8

Substituting 16 with 2^4, we get:

3 * (24)((x^2 - 29) / (-3x)) - 10 * (24)((x^2 - 29) / (-6x)) > 8

Now, we apply the power of a power rule, which states that (am)n = a^(m*n). This rule will allow us to further simplify the exponents:

3 * 2^(4 * (x^2 - 29) / (-3x)) - 10 * 2^(4 * (x^2 - 29) / (-6x)) > 8

Simplifying the exponents, we have:

3 * 2^((-4/3) * (x^2 - 29) / x) - 10 * 2^((-2/3) * (x^2 - 29) / x) > 8

This simplification is a crucial step because it exposes a common exponential term that we can use for substitution. By carefully manipulating the exponents, we have transformed the original expression into a more manageable form. This process highlights the importance of understanding and applying the properties of exponents in solving exponential inequalities. The ability to simplify complex expressions is a fundamental skill in mathematics, and this step demonstrates how it can be effectively used to tackle seemingly challenging problems. As we proceed, we will continue to build upon this foundation, using algebraic techniques to further unravel the inequality and ultimately arrive at the solution.

2. Introduce a Substitution

To make the inequality more tractable, we introduce a substitution. This is a common technique in mathematics, where we replace a complex expression with a single variable to simplify the equation or inequality. In this case, we observe that the terms 2^((-2/3) * (x^2 - 29) / x) and 2^((-4/3) * (x^2 - 29) / x) are related. Specifically, the exponent in the first term is half of the exponent in the second term. This suggests a suitable substitution:

Let t = 2^((-2/3) * (x^2 - 29) / x)

Then, we can express the other exponential term in terms of t:

2^((-4/3) * (x^2 - 29) / x) = (2^((-2/3) * (x^2 - 29) / x))^2 = t^2

Now, we substitute these expressions back into the original inequality:

3 * t^2 - 10 * t > 8

This substitution has transformed the original exponential inequality into a quadratic inequality, which is much easier to solve. This step demonstrates the power of substitution in simplifying complex mathematical expressions. By carefully choosing the substitution, we have effectively reduced the problem to a more familiar form. This is a testament to the importance of strategic thinking in problem-solving. The ability to identify patterns and relationships within an expression is crucial for selecting the appropriate substitution. As we continue to solve the inequality, we will build upon this simplified form, using algebraic techniques to find the values of t that satisfy the inequality. These values will then be used to determine the solutions for x. The substitution technique is a versatile tool in mathematics, and its application in this context highlights its effectiveness in simplifying complex problems.

3. Solve the Quadratic Inequality

Now we have a quadratic inequality: 3t^2 - 10t > 8. To solve this, we first rearrange the inequality to have zero on one side:

3t^2 - 10t - 8 > 0

Next, we factor the quadratic expression:

(3t + 2)(t - 4) > 0

To find the critical points, we set each factor equal to zero:

3t + 2 = 0 => t = -2/3 t - 4 = 0 => t = 4

These critical points divide the number line into three intervals: (-∞, -2/3), (-2/3, 4), and (4, ∞). We test a value from each interval to determine where the inequality is satisfied:

  • Interval (-∞, -2/3): Let t = -1. Then (3(-1) + 2)((-1) - 4) = (-1)(-5) = 5 > 0. So, the inequality holds in this interval.
  • Interval (-2/3, 4): Let t = 0. Then (3(0) + 2)(0 - 4) = (2)(-4) = -8 < 0. So, the inequality does not hold in this interval.
  • Interval (4, ∞): Let t = 5. Then (3(5) + 2)(5 - 4) = (17)(1) = 17 > 0. So, the inequality holds in this interval.

Therefore, the solution for the quadratic inequality in terms of t is:

t < -2/3 or t > 4

This step demonstrates the standard procedure for solving quadratic inequalities. By factoring the quadratic expression and analyzing the sign of the factors in different intervals, we have determined the values of t that satisfy the inequality. This is a fundamental skill in algebra, and its application in this context highlights its importance in solving more complex mathematical problems. The ability to solve quadratic inequalities is essential for a wide range of applications, including optimization problems and modeling real-world scenarios. As we proceed, we will use these solutions for t to determine the solutions for x in the original inequality. This will involve reversing the substitution and solving exponential equations. The process of solving this quadratic inequality is a key step in the overall solution, and it provides a clear illustration of the power of algebraic techniques.

4. Reverse the Substitution and Solve for x

Now we need to reverse the substitution and solve for x. Recall that we substituted t = 2^((-2/3) * (x^2 - 29) / x). We have two inequalities for t:

  1. t < -2/3
  2. t > 4

Let's consider each inequality separately.

Case 1: t < -2/3

Since t = 2^((-2/3) * (x^2 - 29) / x), and exponential functions with a positive base are always positive, t must be positive. Therefore, the inequality t < -2/3 has no solution.

This observation is crucial because it eliminates one of the potential solution sets. It highlights the importance of considering the properties of functions when solving inequalities. Exponential functions with a positive base are always positive, which means that any negative value for t is not possible. This understanding helps us to streamline the solution process and avoid unnecessary calculations. The realization that t < -2/3 has no solution is a key step in narrowing down the possible values of x. As we proceed, we will focus on the other inequality, t > 4, to determine the remaining solutions for x. This process demonstrates the importance of careful analysis and attention to detail in solving mathematical problems. By systematically considering each case and applying the appropriate mathematical principles, we can arrive at a complete and accurate solution.

Case 2: t > 4

Substituting t = 2^((-2/3) * (x^2 - 29) / x), we get:

2^((-2/3) * (x^2 - 29) / x) > 4

Since 4 = 2^2, we can rewrite the inequality as:

2^((-2/3) * (x^2 - 29) / x) > 2^2

Since the base is the same (2 > 1), we can compare the exponents:

(-2/3) * (x^2 - 29) / x > 2

Multiply both sides by -3/2 (and reverse the inequality sign because we're multiplying by a negative number):

(x^2 - 29) / x < -3

Multiply both sides by x. We need to consider two sub-cases: x > 0 and x < 0.

Sub-case 2.1: x > 0

If x > 0, we have:

x^2 - 29 < -3x x^2 + 3x - 29 < 0

To find the roots of the quadratic equation x^2 + 3x - 29 = 0, we use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a) x = (-3 ± √(3^2 - 4(1)(-29))) / (2(1)) x = (-3 ± √(9 + 116)) / 2 x = (-3 ± √125) / 2 x = (-3 ± 5√5) / 2

So, the roots are x_1 = (-3 - 5√5) / 2 and x_2 = (-3 + 5√5) / 2. Since x > 0, we only consider the positive root x_2 = (-3 + 5√5) / 2. The inequality x^2 + 3x - 29 < 0 holds between the roots. Since we are considering x > 0, the solution in this sub-case is:

0 < x < (-3 + 5√5) / 2

Sub-case 2.2: x < 0

If x < 0, we have to reverse the inequality sign when multiplying by x:

x^2 - 29 > -3x x^2 + 3x - 29 > 0

The roots are the same as in the previous sub-case: x_1 = (-3 - 5√5) / 2 and x_2 = (-3 + 5√5) / 2. The inequality x^2 + 3x - 29 > 0 holds outside the roots. Since we are considering x < 0, the solution in this sub-case is:

x < (-3 - 5√5) / 2

5. Combine the Solutions and Consider Domain Restrictions

Combining the solutions from both sub-cases, we have:

x < (-3 - 5√5) / 2 or 0 < x < (-3 + 5√5) / 2

Now, we need to consider the domain restrictions. The original inequality has a term with x in the denominator, so x cannot be 0. Also, the exponent has the term (x^2 - 29) / x. Therefore, x cannot be 0.

Thus, the final solution is:

x < (-3 - 5√5) / 2 or 0 < x < (-3 + 5√5) / 2

Conclusion

In this article, we successfully solved the inequality 3 * 16((x2 - 29) / (-3x)) - 10 * 16((x2 - 29) / (-6x)) > 8. We employed a step-by-step approach, starting with simplifying exponential terms, introducing a substitution to transform the inequality into a quadratic form, solving the quadratic inequality, reversing the substitution, and finally, combining the solutions while considering domain restrictions. This problem highlights the importance of algebraic manipulation, the properties of exponents, and the strategic use of substitution techniques in solving complex mathematical problems. The final solution, x < (-3 - 5√5) / 2 or 0 < x < (-3 + 5√5) / 2, represents the set of all real numbers that satisfy the given inequality. This journey through the solution process not only provides the answer but also reinforces the fundamental principles of mathematical problem-solving. The ability to dissect a complex problem into smaller, manageable steps is a crucial skill that can be applied to a wide range of mathematical challenges. Furthermore, the emphasis on domain restrictions underscores the importance of careful analysis and attention to detail in ensuring the validity of the solution. This article serves as a valuable resource for anyone seeking to enhance their understanding of exponential inequalities and problem-solving strategies in mathematics. The techniques and concepts discussed here can be applied to a variety of similar problems, making this a comprehensive guide to tackling complex inequalities. The journey of solving this inequality is a testament to the power of mathematical reasoning and the elegance of problem-solving strategies. By carefully dissecting the problem and applying the appropriate techniques, we can arrive at a solution that not only satisfies the inequality but also deepens our appreciation for the interconnectedness of mathematical concepts.