Solving Absolute Value Inequalities $5|x+7|+2>8$ Step-by-Step

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In the realm of mathematics, absolute value inequalities present a unique challenge that requires a blend of algebraic manipulation and logical reasoning. Absolute value inequalities, such as the one we'll be dissecting today, involve expressions containing the absolute value of a variable, often accompanied by inequality symbols. These inequalities can be a bit tricky at first glance, but with a systematic approach, they become quite manageable. In this comprehensive guide, we'll not only solve the specific inequality $5|x+7|+2>8$ but also delve into the general strategies and concepts needed to tackle similar problems with confidence. Understanding how to solve these inequalities is a fundamental skill in algebra, serving as a cornerstone for more advanced mathematical concepts. This article aims to provide a step-by-step solution to the given inequality, accompanied by clear explanations and relevant tips. We will break down each step, ensuring that you not only understand the solution but also grasp the underlying principles. So, whether you're a student preparing for an exam, a math enthusiast eager to expand your knowledge, or simply someone looking to brush up on your algebra skills, this guide is tailored to help you master the art of solving absolute value inequalities.

Understanding Absolute Value

Before diving into the solution, let's first revisit the concept of absolute value. The absolute value of a number is its distance from zero on the number line. This distance is always non-negative. Mathematically, the absolute value of a number $x$, denoted as $|x|$, is defined as:

$|x| = x$ if $x ≥ 0$

$|x| = -x$ if $x < 0$

In simpler terms, if $x$ is positive or zero, its absolute value is just $x$ itself. If $x$ is negative, its absolute value is the negation of $x$, which makes it positive. For example, $|5| = 5$ and $|-5| = -(-5) = 5$. The absolute value function essentially strips away the sign of the number, leaving only its magnitude. Understanding this fundamental concept is crucial because absolute value inequalities inherently deal with distances and ranges on the number line. When solving these inequalities, we need to consider both the positive and negative cases of the expression inside the absolute value. This is because the expression inside the absolute value can be either positive or negative, but its distance from zero must satisfy the inequality. For instance, in our given inequality $5|x+7|+2>8$, we need to consider both the case where $x+7$ is positive or zero and the case where $x+7$ is negative. Each case will lead to a different algebraic expression, and solving both will give us the complete solution set for the inequality. This approach of considering different cases is a hallmark of solving absolute value problems and is essential for ensuring that no possible solutions are overlooked.

Solving the Inequality Step-by-Step

Now, let's tackle the inequality $5|x+7|+2>8$ step-by-step. Our primary goal is to isolate the absolute value expression on one side of the inequality. This involves performing algebraic manipulations to simplify the inequality and make it easier to analyze. The first step in this process is to subtract 2 from both sides of the inequality:

$5|x+7|+2-2 > 8-2$

This simplifies to:

$5|x+7| > 6$

Next, we divide both sides by 5 to isolate the absolute value term:

$\frac{5|x+7|}{5} > \frac{6}{5}$

This gives us:

$|x+7| > \frac{6}{5}$

Now that we have isolated the absolute value expression, we can proceed to the next critical step: considering the two possible cases arising from the absolute value. This is where the definition of absolute value comes into play. We need to consider what happens when the expression inside the absolute value, in this case $x+7$, is positive or zero, and what happens when it is negative. This branching into two cases is a standard technique for solving absolute value inequalities and is essential for capturing the complete solution set. By carefully analyzing each case, we can determine the ranges of $x$ that satisfy the original inequality. This methodical approach ensures that we don't miss any solutions and that our final answer is accurate and comprehensive.

Case 1: $x+7 ≥ 0$

In this case, $|x+7| = x+7$. So, the inequality becomes:

$x+7 > \frac{6}{5}$

To solve for $x$, we subtract 7 from both sides:

$x > \frac{6}{5} - 7$

To combine the terms on the right-hand side, we need a common denominator. Since 7 can be written as $\frac{35}{5}$, we have:

$x > \frac{6}{5} - \frac{35}{5}$

$x > \frac{6-35}{5}$

$x > \frac{-29}{5}$

This gives us one part of our solution: $x$ must be greater than $-\frac{29}{5}$ in this case. However, we must also remember the initial condition for this case: $x+7 ≥ 0$. Let's solve this inequality to find the relevant range of $x$:

$x+7 ≥ 0$

$x ≥ -7$

Now, we need to consider the intersection of the two conditions we found in this case: $x > -\frac{29}{5}$ and $x ≥ -7$. Since $-\frac{29}{5}$ is approximately -5.8, and -5.8 is greater than -7, the stricter condition is $x > -\frac{29}{5}$. This means that for this case, the solutions are all values of $x$ that are greater than $-\frac{29}{5}$. It's crucial to consider both the inequality derived from the absolute value and the initial condition for the case to ensure that the solutions we find are valid within the context of the case we're analyzing. This careful consideration of conditions and intersections is a key aspect of solving absolute value inequalities accurately.

Case 2: $x+7 < 0

In this case, $|x+7| = -(x+7)$. So, the inequality becomes:

$-(x+7) > \frac{6}{5}$

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

$x+7 < -\frac{6}{5}$

Now, we subtract 7 from both sides:

$x < -\frac{6}{5} - 7$

Again, we need a common denominator to combine the terms on the right-hand side:

$x < -\frac{6}{5} - \frac{35}{5}$

$x < -\frac{6+35}{5}$

$x < -\frac{41}{5}$

This gives us another part of our solution: $x$ must be less than $-\frac{41}{5}$ in this case. We also need to consider the initial condition for this case: $x+7 < 0$. Let's solve this inequality:

$x+7 < 0$

$x < -7$

Now, we consider the intersection of the two conditions we found in this case: $x < -\frac{41}{5}$ and $x < -7$. Since $-\frac{41}{5}$ is -8.2, and -8.2 is less than -7, the stricter condition is $x < -\frac{41}{5}$. This means that for this case, the solutions are all values of $x$ that are less than $-\frac{41}{5}$. Just as in Case 1, considering both the inequality derived from the absolute value and the initial condition is vital for ensuring the validity of our solutions. By carefully analyzing the conditions and their intersections, we can confidently determine the correct solution set for each case.

Combining the Solutions

Now that we have solved both cases, we need to combine the solutions to find the overall solution set for the original inequality $5|x+7|+2>8$. From Case 1, we found that $x > -\frac{29}{5}$, and from Case 2, we found that $x < -\frac{41}{5}$. These two solution sets represent intervals on the number line. The first interval is all numbers greater than $-\frac{29}{5}$, and the second interval is all numbers less than $-\frac{41}{5}$. Since these are separate cases, we combine their solutions using the union symbol ($\cup$). This means that the overall solution set consists of all numbers that satisfy either one of the cases.

Therefore, the solution set is:

$x < -\frac{41}{5}$ or $x > -\frac{29}{5}$

In interval notation, this is written as:

$(-\infty, -\frac{41}{5}) \cup (-\frac{29}{5}, \infty)$

This final answer represents all the values of $x$ that make the original inequality true. It's crucial to express the solution set clearly and accurately, whether in inequality notation or interval notation. In this case, the interval notation provides a concise and unambiguous way to represent the range of values that satisfy the inequality. By combining the solutions from both cases, we have successfully solved the absolute value inequality and identified the complete set of values that fulfill the given condition.

Final Answer

Comparing our solution with the given form $(A, B)$ and $(-\infty, A) \cup (B, \infty)$, we can identify the values of A and B.

$A = -\frac{41}{5}$

$B = -\frac{29}{5}$

Therefore, the final answer is:

$\boxed{\left(-\infty, -\frac{41}{5}\right) \cup \left(-\frac{29}{5}, \infty\right)}$

This solution set represents all real numbers that satisfy the inequality $5|x+7|+2>8$. By following a systematic approach, considering both cases arising from the absolute value, and carefully combining the solutions, we have successfully navigated this problem. Remember that practice is key to mastering these types of inequalities. By working through various examples and understanding the underlying principles, you can build confidence and proficiency in solving absolute value inequalities.