Solving Absolute Value Inequality |(1/3)x + 1| < 8 A Step-by-Step Guide

In this comprehensive guide, we will delve into the intricacies of solving absolute value inequalities, specifically addressing the problem: $\left|\frac{1}{3} x+1\right|<8$. Absolute value inequalities often present a challenge to students and require a systematic approach to solve them accurately. Understanding the fundamental properties of absolute values and inequalities is crucial for tackling these types of problems. We will break down the problem step by step, explaining the underlying concepts and techniques involved. This guide aims to provide a clear and concise understanding, enabling you to solve similar problems with confidence.

Absolute value represents the distance of a number from zero on the number line. Therefore, when we deal with an absolute value inequality like $\left|\frac{1}{3} x+1\right|<8$, we are essentially looking for all values of $x$ for which the distance of $\frac{1}{3} x+1$ from zero is less than 8. This concept leads to two separate inequalities that need to be considered simultaneously. One inequality represents the case where the expression inside the absolute value is positive or zero, and the other represents the case where the expression inside the absolute value is negative. By solving both inequalities, we can determine the range of values for $x$ that satisfy the original absolute value inequality.

To effectively solve absolute value inequalities, it's essential to grasp the core principles that govern them. The absolute value of a number, denoted as $|a|$, is its non-negative magnitude, irrespective of its sign. For instance, $|5|$ and $|-5|$ both equal 5. When solving inequalities involving absolute values, this property necessitates the consideration of two distinct scenarios: one where the expression inside the absolute value is positive or zero, and another where it is negative. This bifurcation stems from the fact that both a positive and a negative value within the absolute value can satisfy the inequality. For example, if we have $|x| < 3$, this implies that $x$ must be both less than 3 and greater than -3, since both positive and negative values within this range have an absolute value less than 3. This dual consideration is the cornerstone of solving absolute value inequalities and requires a methodical approach to ensure all possible solutions are captured.

Understanding the Inequality

The given inequality is $\left|\frac{1}{3} x+1\right|<8$. This inequality states that the absolute value of the expression $\frac{1}{3} x+1$ must be less than 8. In simpler terms, the distance of $\frac{1}{3} x+1$ from zero must be less than 8. This condition implies that $\frac{1}{3} x+1$ must lie between -8 and 8. We can express this as a compound inequality, which is a crucial step in solving absolute value inequalities. The ability to translate the absolute value inequality into a compound inequality is fundamental to finding the solution set.

The concept of absolute value as a distance from zero is central to understanding and solving these types of inequalities. When an expression inside the absolute value bars is less than a certain positive number, it means that the expression's value must fall within a specific range centered around zero. For example, if $|y| < 5$, then $y$ must be between -5 and 5. This is because any value of $y$ outside this range would have an absolute value greater than or equal to 5. Visualizing this on a number line can be particularly helpful. Imagine a number line with zero at the center. The inequality $|y| < 5$ represents all points on the number line that are less than 5 units away from zero. These points will lie between -5 and 5, not including the endpoints. This understanding forms the basis for converting absolute value inequalities into compound inequalities and ultimately solving for the unknown variable.

Splitting into Two Inequalities

To solve this absolute value inequality, we need to split it into two separate inequalities. This is because the absolute value function has two possible cases: the expression inside the absolute value can be positive or negative. When the expression is positive, the absolute value is simply the expression itself. When the expression is negative, the absolute value is the negation of the expression. This leads to two inequalities:

1. $\frac{1}{3} x+1 < 8$
2. $-\left(\frac{1}{3} x+1\right) < 8$

The first inequality, $\frac{1}{3} x+1 < 8$, represents the case where the expression inside the absolute value, $\frac{1}{3} x+1$, is positive or zero. In this case, the absolute value simply removes the absolute value bars, and we can proceed to solve the inequality as usual. The second inequality, $-\left(\frac{1}{3} x+1\right) < 8$, represents the case where the expression $\frac{1}{3} x+1$ is negative. In this case, the absolute value effectively negates the expression, making it positive. Therefore, we need to consider the inequality with the negated expression. Solving these two inequalities will give us the range of values for $x$ that satisfy the original absolute value inequality. It's crucial to remember that both inequalities must be considered to obtain the complete solution set.

Solving the First Inequality

Let's solve the first inequality: $\frac{1}{3} x+1 < 8$. To isolate $x$, we first subtract 1 from both sides:

$\frac{1}{3} x < 8 - 1$ $\frac{1}{3} x < 7$

Next, we multiply both sides by 3 to get rid of the fraction:

$x < 7 \times 3$ $x < 21$

So, the solution to the first inequality is $x < 21$. This means that any value of $x$ less than 21 satisfies the first part of our absolute value inequality. However, we also need to consider the second inequality to find the complete solution set. The step-by-step process of isolating $x$ is a fundamental algebraic technique that is essential for solving a wide range of inequalities and equations. By performing the same operation on both sides of the inequality, we maintain the balance and progressively simplify the expression until we arrive at the solution. In this case, subtracting 1 from both sides and then multiplying by 3 allowed us to isolate $x$ and determine the upper bound of the solution set for the first inequality.

Solving the Second Inequality

Now, let's solve the second inequality: $-\left(\frac{1}{3} x+1\right) < 8$. First, we distribute the negative sign:

$-\frac{1}{3} x - 1 < 8$

Next, we add 1 to both sides:

$-\frac{1}{3} x < 8 + 1$ $-\frac{1}{3} x < 9$

Now, we multiply both sides by -3. Remember, when we multiply or divide an inequality by a negative number, we must flip the inequality sign:

$x > 9 \times -3$ $x > -27$

So, the solution to the second inequality is $x > -27$. This means that any value of $x$ greater than -27 satisfies the second part of our absolute value inequality. Combining this with the solution from the first inequality will give us the complete solution set for the original problem. The critical step in this process is remembering to flip the inequality sign when multiplying or dividing by a negative number. This is a common point of error for students, so it is crucial to pay close attention to the sign and its effect on the direction of the inequality.

Combining the Solutions

We found that $x < 21$ and $x > -27$. Combining these two inequalities, we get the solution range for $x$: $-27 < x < 21$. This means that $x$ must be greater than -27 and less than 21 to satisfy the original absolute value inequality. This range represents all the values of $x$ that make the expression $\left|\frac{1}{3} x+1\right|$ less than 8. We can visualize this solution on a number line, where the interval between -27 and 21 is shaded, indicating that all points within this interval are solutions. The endpoints -27 and 21 are not included in the solution set because the inequality is strict (i.e., less than, not less than or equal to). Combining the solutions from the two separate inequalities is a crucial step in solving absolute value inequalities. It is essential to understand that the solution set is the intersection of the solutions from each individual inequality, representing the values of $x$ that satisfy both conditions simultaneously.

Final Answer

The solution to the inequality $\left|\frac{1}{3} x+1\right|<8$ is $-27 < x < 21$. This corresponds to the interval $(-27, 21)$. In the given options, this can be expressed as: B. $-2721$ (Simplify your answers.)

In conclusion, solving absolute value inequalities requires a systematic approach that involves splitting the inequality into two separate cases and solving each one individually. Remember to consider both the positive and negative scenarios for the expression inside the absolute value. By carefully following the steps outlined in this guide, you can confidently solve a wide range of absolute value inequalities. Understanding the underlying concepts and practicing regularly will enhance your problem-solving skills and ensure accuracy. Absolute value inequalities are a fundamental topic in algebra, and mastering them will provide a solid foundation for more advanced mathematical concepts. This comprehensive guide has aimed to provide a clear and concise understanding of the topic, empowering you to tackle similar problems with ease and confidence. Keep practicing, and you'll become proficient in solving absolute value inequalities.