Solving 5|a-1|+6=21 A Step-by-Step Guide

In mathematics, absolute value equations often present a unique challenge due to the nature of the absolute value function. The absolute value of a number is its distance from zero, which means it is always non-negative. When solving equations involving absolute values, it is crucial to consider both positive and negative possibilities. This article delves into the step-by-step solution of the equation 5|a-1|+6=21, exploring the underlying principles and techniques involved. Our primary focus here is to demystify the process of solving absolute value equations, providing a comprehensive guide that will benefit students and math enthusiasts alike. Understanding how to tackle these equations is a fundamental skill in algebra, and this article aims to make that skill accessible and straightforward.

Before diving into the specific equation, it's essential to grasp the concept of absolute value. The absolute value of a number, denoted as |x|, represents its distance from zero on the number line. This means that |x| is always non-negative. For example, |3| = 3 and |-3| = 3. This property leads to a critical consideration when solving equations involving absolute values: there are typically two cases to consider, the positive and the negative. This dual nature of absolute value is what makes solving such equations a bit more intricate than standard algebraic equations. Failing to account for both cases can lead to incomplete or incorrect solutions. The absolute value is a cornerstone concept in various mathematical fields, including calculus, analysis, and complex numbers, making its thorough understanding vital for anyone serious about mathematics. By mastering absolute value, one can tackle more complex problems with greater confidence and accuracy.

To solve the equation 5|a-1|+6=21, we will proceed in a systematic manner, breaking down each step for clarity. This methodical approach ensures that we don't miss any potential solutions and fully understand the process. First, it's crucial to isolate the absolute value term on one side of the equation. This is akin to peeling away the layers to get to the core of the problem. Once the absolute value is isolated, we can then consider the two possible cases arising from the definition of absolute value. Let’s break down the steps:

1. Isolate the Absolute Value Term:

   • Begin by subtracting 6 from both sides of the equation:
     • 5|a-1| + 6 - 6 = 21 - 6
     • This simplifies to:
     • 5|a-1| = 15
   • Next, divide both sides by 5 to isolate the absolute value:
     • (5|a-1|)/5 = 15/5
     • This gives us:
     • |a-1| = 3

2. Consider the Two Cases:

   • Case 1: The expression inside the absolute value is positive or zero:

     • If a-1 is positive or zero, then |a-1| = a-1. Thus,
     • a - 1 = 3

   • Case 2: The expression inside the absolute value is negative:

     • If a-1 is negative, then |a-1| = -(a-1). Thus,
     • -(a - 1) = 3

3. Solve Each Case:

   • Solving Case 1:

     • To solve a - 1 = 3, add 1 to both sides:
     • a - 1 + 1 = 3 + 1
     • Which gives us:
     • a = 4

   • Solving Case 2:

     • To solve -(a - 1) = 3, first, distribute the negative sign:
     • -a + 1 = 3
     • Subtract 1 from both sides:
     • -a + 1 - 1 = 3 - 1
     • This simplifies to:
     • -a = 2
     • Multiply both sides by -1 to solve for a:
     • (-1)(-a) = (-1)(2)
     • Which gives us:
     • a = -2

4. Final Solutions:

   • Therefore, the solutions to the equation 5|a-1|+6=21 are a = 4 and a = -2. It's always good practice to check these solutions in the original equation to ensure they are correct. Substituting these values back into the original equation confirms their validity.

To ensure the accuracy of our solutions, it is imperative to verify them by substituting them back into the original equation. This step is not just a formality; it's a crucial part of the problem-solving process. Verification helps to catch any potential errors made during the solution steps and solidifies our confidence in the final answer. Moreover, this practice reinforces the understanding of the equation and the behavior of absolute values. In this section, we will meticulously check both solutions we obtained, a = 4 and a = -2, against the original equation, 5|a-1|+6=21. This process will not only confirm our answers but also enhance our grasp of the equation's properties and solutions.

• Verification for a = 4:

  • Substitute a = 4 into the original equation:
    • 5|4-1| + 6 = 21
    • Simplify the expression inside the absolute value:
    • 5|3| + 6 = 21
    • Compute the absolute value:
    • 5(3) + 6 = 21
    • Perform the multiplication:
    • 15 + 6 = 21
    • Finally, add the numbers:
    • 21 = 21
  • Since the equation holds true, a = 4 is indeed a valid solution.

• Verification for a = -2:

  • Now, substitute a = -2 into the original equation:
    • 5|-2-1| + 6 = 21
    • Simplify the expression inside the absolute value:
    • 5|-3| + 6 = 21
    • Compute the absolute value:
    • 5(3) + 6 = 21
    • Perform the multiplication:
    • 15 + 6 = 21
    • Finally, add the numbers:
    • 21 = 21
  • Again, the equation holds true, confirming that a = -2 is also a valid solution.

While the step-by-step method described above is a standard approach, exploring alternative methods can deepen our understanding of absolute value equations and provide additional tools for problem-solving. One such approach involves graphical representation, which can offer a visual understanding of the solutions. Another method involves recognizing patterns and applying properties of absolute values to simplify the equation. Additionally, using technology, such as graphing calculators or mathematical software, can help visualize solutions and verify results. This section will briefly discuss these alternative methods, encouraging a more versatile and comprehensive approach to solving absolute value equations. By familiarizing ourselves with various techniques, we become more adept at tackling a wide range of problems and gain a more intuitive sense of mathematical concepts.

• Graphical Method:

  • One way to visualize the solutions of the equation 5|a-1|+6=21 is to treat each side of the equation as a separate function and graph them. Let y1 = 5|a-1|+6 and y2 = 21. The solutions to the equation are the x-coordinates (in this case, 'a'-coordinates) of the points where the two graphs intersect. Graphing these functions would visually show the two points of intersection at a = 4 and a = -2, confirming our solutions.

• Using Properties of Absolute Value:

  • Another approach involves applying the properties of absolute value directly. After isolating the absolute value term, we have |a-1| = 3. This equation implies that the distance between 'a' and 1 is 3 units. We can think of this on a number line: what numbers are 3 units away from 1? This leads us directly to a = 4 (1 + 3) and a = -2 (1 - 3). This method can be quicker for simpler equations and reinforces the geometric interpretation of absolute value.

• Technology-Assisted Solutions:

  • Graphing calculators and mathematical software (like Desmos, GeoGebra, or Wolfram Alpha) can be used to solve absolute value equations. By inputting the equation or the two functions (as in the graphical method), these tools can display the solutions or the points of intersection, respectively. This can be especially helpful for more complex equations or when verifying solutions obtained by hand. However, it's crucial to understand the underlying mathematical principles rather than relying solely on technology.

When solving absolute value equations, certain common mistakes can lead to incorrect answers. Being aware of these pitfalls can significantly improve accuracy and problem-solving efficiency. One frequent error is forgetting to consider both the positive and negative cases of the absolute value. Another mistake is incorrectly isolating the absolute value term before splitting the equation into cases. Additionally, arithmetic errors during the solution process can lead to incorrect results. It's essential to double-check each step and carefully apply the correct algebraic manipulations. This section highlights these common mistakes and provides tips to avoid them, ensuring a more successful approach to solving absolute value equations. By recognizing and preventing these errors, we can enhance our mathematical skills and achieve more reliable solutions.

• Forgetting the Negative Case:

  • As emphasized earlier, the most common mistake is neglecting to consider both positive and negative cases for the expression inside the absolute value. Remember that |x| = a implies that x = a or x = -a. Failing to account for both possibilities will result in only one solution, leading to an incomplete answer. Always explicitly write out both cases to avoid this error.

• Incorrectly Isolating the Absolute Value:

  • Before splitting the equation into two cases, the absolute value term must be completely isolated on one side of the equation. Operations like addition, subtraction, multiplication, or division should be performed to ensure that the absolute value expression is by itself. For instance, in the equation 5|a-1|+6=21, you must subtract 6 and divide by 5 before considering the cases a-1 = 3 and a-1 = -3. Skipping or misapplying this step will lead to incorrect solutions.

• Arithmetic Errors:

  • As with any algebraic problem, arithmetic errors can creep in during the solution process. These can range from simple addition or subtraction mistakes to errors in multiplication or division. To minimize these errors, work carefully, show your steps clearly, and double-check your calculations. If possible, use a calculator to verify arithmetic operations, especially when dealing with more complex numbers.

• Incorrectly Distributing the Negative Sign:

  • When dealing with the negative case, such as -(a-1) = 3, it's crucial to distribute the negative sign correctly. This means multiplying each term inside the parentheses by -1. A common mistake is to apply the negative sign only to the first term, resulting in -a - 1 = 3 instead of -a + 1 = 3. Always ensure that the negative sign is properly distributed to all terms within the parentheses.

Absolute value equations are not just abstract mathematical concepts; they have practical applications in various real-world scenarios. These equations are particularly useful in situations where we are concerned with the magnitude or distance of a quantity, regardless of its direction or sign. For instance, absolute values are used in engineering to calculate tolerances, in physics to describe motion, and in finance to model deviations from a target value. Understanding these applications can help appreciate the relevance and importance of mastering absolute value equations. This section will explore some specific examples of how these equations are used in real-world contexts, demonstrating their practical utility and relevance.

• Engineering:

  • In engineering, absolute value equations are used to define tolerances or acceptable deviations in measurements. For example, when manufacturing parts, a certain dimension might have a specified tolerance range. If a part is designed to be 10 cm long with a tolerance of ±0.1 cm, this can be expressed as |length - 10| ≤ 0.1. This equation ensures that the manufactured part's length is within the acceptable range, regardless of whether it's slightly shorter or longer than the target dimension.

• Physics:

  • In physics, absolute value is often used to describe the magnitude of a vector quantity, such as velocity or acceleration, without regard to its direction. For example, the speed of an object is the absolute value of its velocity. If an object's velocity is described by the function v(t), then its speed is |v(t)|. Similarly, absolute value can be used to calculate the distance traveled by an object, regardless of its direction of movement.

• Finance:

  • In finance, absolute value equations can be used to model deviations from a target or expected value. For example, if an investor aims for a certain return on investment (ROI) of 8%, the absolute value can be used to express the acceptable range of deviation from this target. The equation |actual ROI - 8| ≤ 2 might represent that the actual ROI should be within 2 percentage points of the target, either above or below. This helps in assessing the risk and performance of investments.

• Error Analysis:

  • In various scientific and engineering fields, absolute value is used in error analysis to quantify the magnitude of errors without considering their sign. For example, if an experiment's result is expected to be 5.0 units, and the actual result is 4.8 units, the absolute error is |4.8 - 5.0| = 0.2 units. This provides a clear measure of the error's size, irrespective of whether the measured value is higher or lower than the expected value.

In summary, solving the absolute value equation 5|a-1|+6=21 involves a systematic approach that includes isolating the absolute value term, considering both positive and negative cases, solving each case independently, and verifying the solutions. We found that the solutions to the equation are a = 4 and a = -2. This process highlights the importance of understanding the definition of absolute value and its implications for solving equations. Furthermore, we explored alternative methods such as graphical representation and the use of absolute value properties, which can enhance problem-solving skills. By avoiding common mistakes and appreciating real-world applications, one can gain a deeper understanding and proficiency in solving absolute value equations. This skill is not only crucial in mathematics but also in various fields that rely on quantitative analysis and modeling.

The ability to solve absolute value equations is a fundamental skill in algebra and has far-reaching applications. Mastering this topic requires a solid grasp of the underlying principles, a careful approach to problem-solving, and consistent practice. The step-by-step methods, alternative approaches, and error-avoidance tips discussed in this article provide a comprehensive guide for learners at various levels. As we have seen, absolute value equations are more than just abstract mathematical expressions; they are powerful tools for modeling and solving real-world problems. By continuing to explore and apply these concepts, we can strengthen our mathematical abilities and enhance our understanding of the world around us.