Solve The System Of Inequalities: Log₂(x-4)² ≤ 2 And (x-1)² > 4

In the realm of mathematics, solving systems of inequalities is a fundamental skill with widespread applications across various fields, including economics, engineering, and computer science. This article delves into the intricacies of solving a specific system of inequalities, providing a step-by-step guide to navigate the process effectively. We will explore the underlying concepts, techniques, and potential pitfalls, equipping you with the knowledge and confidence to tackle similar problems with ease. Let's embark on this mathematical journey and unravel the solution to the given system of inequalities.

The key to successfully solving systems of inequalities lies in a methodical approach, breaking down the problem into manageable steps and applying the appropriate techniques. By carefully analyzing each inequality, identifying critical points, and considering the domain of the variables, we can arrive at the solution set that satisfies all the given conditions. This article aims to provide a comprehensive understanding of this process, empowering you to confidently tackle a wide range of inequality problems.

When we are presented with a system of inequalities, our primary goal is to find the set of values that satisfy all inequalities simultaneously. This means identifying the range of values for the variable(s) that make each inequality true. To achieve this, we typically follow a systematic approach that involves solving each inequality individually, identifying critical points, and then combining the solutions to determine the overall solution set. The process may vary depending on the complexity of the inequalities, but the underlying principles remain the same.

Let's consider the specific system of inequalities presented:

$$\begin{cases} log_2(x-4)^2 \le 2 \\ (x-1)^2 > 4 \end{cases}$$

This system comprises two inequalities: a logarithmic inequality and a quadratic inequality. To solve this system, we will address each inequality separately and then combine their solutions to find the common solution set. This step-by-step approach will allow us to break down the problem into smaller, more manageable parts, making the overall solution process more transparent and less prone to errors. We will meticulously analyze each inequality, paying close attention to the domain restrictions and potential pitfalls, to ensure that we arrive at the correct solution.

Solving the Logarithmic Inequality

Let's begin by tackling the first inequality:

$$log_2(x-4)^2 \le 2$$

To solve this logarithmic inequality, we need to understand the properties of logarithms and how they interact with inequalities. The fundamental principle here is to rewrite the inequality in exponential form, which will allow us to eliminate the logarithm and work with a more familiar algebraic expression. We must also consider the domain of the logarithmic function, ensuring that the argument of the logarithm remains positive.

First, we can rewrite the inequality in exponential form. Recall that the logarithmic expression $log_b(a) = c$ is equivalent to the exponential expression $b^c = a$. Applying this principle to our inequality, we get:

$$(x-4)^2 \le 2^2$$

This transformation allows us to remove the logarithm and work with a simpler algebraic expression. However, before we proceed further, it is crucial to consider the domain of the logarithmic function. The argument of the logarithm, $(x-4)^2$, must be strictly greater than zero. This is because logarithms are only defined for positive arguments. Therefore, we have the following condition:

$$(x-4)^2 > 0$$

This condition implies that $x-4 \ne 0$, which means $x \ne 4$. This restriction is essential and must be considered when we combine the solutions later. Failing to account for the domain restriction can lead to extraneous solutions or an incorrect solution set. Now, let's proceed with solving the squared inequality:

$$(x-4)^2 \le 4$$

Taking the square root of both sides, we get:

$$|x-4| \le 2$$

This absolute value inequality can be rewritten as two separate inequalities:

$$-2 \le x-4 \le 2$$

Adding 4 to all parts of the inequality, we have:

$$2 \le x \le 6$$

This interval represents the solution to the squared inequality. However, we must remember the domain restriction we identified earlier: $x \ne 4$. Therefore, the solution to the logarithmic inequality is the interval $[2, 6]$ excluding the point $x=4$. This can be expressed as:

$$x \in [2, 4) \cup (4, 6]$$

This is a crucial step in the solution process. We have successfully transformed the logarithmic inequality into a more manageable algebraic form, considered the domain restriction, and arrived at a solution set that satisfies the original inequality. Now, we move on to the next inequality in the system.

Solving the Quadratic Inequality

Next, let's solve the second inequality:

$$(x-1)^2 > 4$$

This is a quadratic inequality, and to solve it, we can follow a similar approach as before: expand the square, rearrange the terms, and then find the roots. The roots will help us identify the intervals where the inequality holds true. Let's begin by expanding the square:

$$x^2 - 2x + 1 > 4$$

Now, subtract 4 from both sides to get a quadratic expression on one side and zero on the other:

$$x^2 - 2x - 3 > 0$$

To solve this inequality, we first find the roots of the corresponding quadratic equation:

$$x^2 - 2x - 3 = 0$$

We can factor this quadratic equation as follows:

$$(x-3)(x+1) = 0$$

This gives us two roots: $x = 3$ and $x = -1$. These roots are the critical points that divide the number line into intervals. Now, we need to determine the intervals where the inequality $x^2 - 2x - 3 > 0$ holds true. To do this, we can test a value from each interval in the inequality.

The intervals are: $(-\infty, -1)$, $(-1, 3)$, and $(3, \infty)$. Let's test a value from each interval:

• For the interval $(-\infty, -1)$, let's choose $x = -2$:

  $$(-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5 > 0$$

  The inequality holds true in this interval.
• For the interval $(-1, 3)$, let's choose $x = 0$:

  $$(0)^2 - 2(0) - 3 = -3 < 0$$

  The inequality does not hold true in this interval.
• For the interval $(3, \infty)$, let's choose $x = 4$:

  $$(4)^2 - 2(4) - 3 = 16 - 8 - 3 = 5 > 0$$

  The inequality holds true in this interval.

Therefore, the solution to the quadratic inequality $(x-1)^2 > 4$ is:

$$x \in (-\infty, -1) \cup (3, \infty)$$

We have successfully solved the quadratic inequality and identified the intervals where the inequality holds true. Now, we have the solutions to both inequalities in the system. The final step is to combine these solutions to find the common solution set.

Combining the Solutions

Now that we have solved both inequalities individually, we need to find the values of $x$ that satisfy both inequalities simultaneously. This means finding the intersection of the solution sets we obtained in the previous steps. The solution to the first inequality, $log_2(x-4)^2 \le 2$, is:

$$x \in [2, 4) \cup (4, 6]$$

The solution to the second inequality, $(x-1)^2 > 4$, is:

$$x \in (-\infty, -1) \cup (3, \infty)$$

To find the intersection of these two solution sets, we can visualize them on a number line. The first solution set includes the interval from 2 to 6, excluding 4. The second solution set includes the intervals from negative infinity to -1 and from 3 to positive infinity. The intersection of these sets will be the regions where both solutions overlap.

Upon careful examination, we can see that the overlapping regions are:

• The interval $[2, 4)$ intersects with the interval $(3, \infty)$, resulting in the interval $(3, 4)$.
• The interval $(4, 6]$ intersects with the interval $(3, \infty)$, resulting in the interval $(4, 6]$.
• There is no overlap between $[2, 4) \cup (4, 6]$ and $(-\infty, -1)$.

Therefore, the final solution to the system of inequalities is the union of these overlapping intervals:

$$x \in (3, 4) \cup (4, 6]$$

This is the set of all values of $x$ that satisfy both the logarithmic and quadratic inequalities. We have successfully combined the solutions of the individual inequalities to arrive at the solution to the entire system.

In this article, we have explored the process of solving a system of inequalities, focusing on a specific example involving a logarithmic inequality and a quadratic inequality. We have demonstrated a step-by-step approach that involves solving each inequality individually, considering domain restrictions, and then combining the solutions to find the common solution set. This methodical approach is crucial for tackling complex inequality problems effectively.

Solving inequalities often requires a combination of algebraic manipulation, understanding of function properties, and careful consideration of domain restrictions. By breaking down the problem into smaller, manageable steps, we can minimize the chances of errors and gain a deeper understanding of the underlying concepts. The techniques and principles discussed in this article can be applied to a wide range of inequality problems, empowering you to confidently tackle mathematical challenges in various contexts.

The solution to the system of inequalities:

$$\begin{cases} log_2(x-4)^2 \le 2 \\ (x-1)^2 > 4 \end{cases}$$

is:

$$x \in (3, 4) \cup (4, 6]$$

This solution represents the set of all real numbers that satisfy both inequalities simultaneously. The process of solving this system has highlighted the importance of considering domain restrictions, algebraic manipulation, and combining solutions effectively. By mastering these skills, you can confidently navigate the world of inequalities and apply them to solve real-world problems.

Mastering the art of solving inequalities is not just about finding the correct answer; it is about developing a logical and systematic approach to problem-solving. The skills you acquire in this process will be invaluable in your mathematical journey and in various other fields that require analytical thinking and problem-solving abilities. So, embrace the challenge, practice diligently, and let the world of inequalities unfold before you.