Solving |x+5|=x+5 A Step-by-Step Guide With Number Line Representation

In the realm of mathematics, absolute value equations often present a unique challenge that requires a comprehensive understanding of the properties of absolute values. At its core, the absolute value of a number represents its distance from zero on the number line, irrespective of direction. This fundamental concept translates into a piecewise definition, where |x| equals x if x is non-negative, and -x if x is negative. This duality is critical when solving equations involving absolute values, as it necessitates considering both scenarios to capture all possible solutions.

When faced with an equation such as |x+5|=x+5, we embark on a journey that intertwines algebraic manipulation with a deep appreciation for the nature of absolute values. The very structure of the equation hints at an underlying condition that must be satisfied for the equality to hold true. On one side, we have the absolute value expression, which by definition is non-negative. On the other side, we have a linear expression, x+5, which can take on any real value. The equation dictates that the distance of x+5 from zero must be equal to the value of x+5 itself. This crucial observation forms the bedrock of our solution strategy.

To effectively tackle this equation, we employ a two-pronged approach, carefully dissecting the implications of the absolute value. First, we consider the scenario where the expression inside the absolute value, x+5, is non-negative. In this case, the absolute value simply vanishes, and we are left with a straightforward linear equation. Second, we explore the scenario where x+5 is negative, necessitating a sign change before we can proceed. By meticulously analyzing these two cases, we ensure that we leave no stone unturned in our quest for the complete solution set. This methodical approach not only leads us to the correct answers but also reinforces our grasp of the fundamental principles governing absolute value equations.

When the expression inside the absolute value, x+5, is greater than or equal to zero, we can directly remove the absolute value signs without any alteration. This stems from the fundamental definition of absolute value, which dictates that the absolute value of a non-negative number is the number itself. In mathematical terms, if x+5≥0, then |x+5|=x+5. This simplification is a cornerstone in solving absolute value equations, as it transforms a potentially complex problem into a more manageable linear equation.

The inequality x+5≥0 serves as a crucial condition that must be satisfied for this case to hold true. By subtracting 5 from both sides, we arrive at the condition x≥-5. This inequality defines a range of x-values for which our assumption of non-negativity is valid. It is paramount to remember this condition, as any solution we obtain in this case must adhere to this constraint. Failing to do so would lead to extraneous solutions that do not satisfy the original equation.

Now, with the absolute value removed, we are left with the equation x+5=x+5. This equation might seem peculiar at first glance, as both sides are identical. However, this is a powerful indication of a specific type of solution. Subtracting x from both sides, we are left with 5=5, a statement that is always true, irrespective of the value of x. This identity implies that any x-value that satisfies the condition x≥-5 will indeed be a solution to the original absolute value equation. This is a significant finding, as it reveals a continuous range of solutions rather than isolated points.

In summary, for the case where x+5 is non-negative, we have established that all x-values greater than or equal to -5 satisfy the equation |x+5|=x+5. This forms a crucial part of the solution set, which we will later combine with the solutions obtained from the second case to construct the complete solution.

The second critical scenario in solving the absolute value equation |x+5|=x+5 arises when the expression inside the absolute value, x+5, is strictly negative. In this case, the absolute value operation introduces a crucial sign change. According to the definition of absolute value, if x+5<0, then |x+5|=-(x+5). This sign change is essential because it ensures that the absolute value, which represents a distance, is always non-negative.

The inequality x+5<0 sets the stage for this case, defining the range of x-values for which our assumption of negativity holds. Subtracting 5 from both sides yields the condition x<-5. This inequality is a critical constraint, as any solution derived in this case must strictly adhere to it. Solutions that do not satisfy this condition are extraneous and must be discarded.

When x+5 is negative, the equation |x+5|=x+5 transforms into -(x+5)=x+5. This transformation is a direct application of the absolute value definition for negative arguments. To solve this equation, we first distribute the negative sign on the left side, resulting in -x-5=x+5. This step is crucial as it eliminates the parentheses and prepares the equation for further simplification.

Next, we aim to isolate the variable x. By adding x to both sides, we obtain -5=2x+5. Subsequently, subtracting 5 from both sides gives us -10=2x. Finally, dividing both sides by 2 yields x=-5. This appears to be a potential solution, but we must scrutinize it against the condition x<-5 that governs this case.

Upon careful examination, we find that x=-5 does not satisfy the condition x<-5. The value -5 is equal to -5, but it is not strictly less than -5. This discrepancy indicates that x=-5 is an extraneous solution in this case. It arises from the sign change introduced by the absolute value and does not truly satisfy the original equation within the context of this case.

Therefore, in the scenario where x+5 is negative, we find that there are no valid solutions to the equation |x+5|=x+5. The apparent solution x=-5 is extraneous and must be rejected. This result is a crucial piece of the puzzle, as it complements the solutions we obtained in the first case to form the complete solution set.

Having meticulously analyzed both cases, we now stand at the pivotal juncture of synthesizing our findings to construct the complete solution set for the equation |x+5|=x+5. In the first case, where x+5 is non-negative (x≥-5), we discovered that all x-values within this range satisfy the equation. This constitutes a continuous interval of solutions extending from -5 to positive infinity.

Conversely, in the second case, where x+5 is negative (x<-5), our investigation revealed that there are no valid solutions. The apparent solution x=-5 was found to be extraneous, leaving us with an empty solution set for this scenario. This absence of solutions in the second case is equally crucial, as it reinforces the completeness of our solution obtained in the first case.

To obtain the final solution set, we combine the solutions from both cases. Since the second case yielded no solutions, the solution set is solely determined by the first case. Therefore, the solution set for the equation |x+5|=x+5 is the interval of all x-values greater than or equal to -5. In mathematical notation, this is expressed as x≥-5.

This solution set encapsulates all real numbers that, when substituted into the original equation, render it a true statement. It is a testament to the power of methodical problem-solving, where we dissect a complex problem into manageable cases, analyze each case independently, and then synthesize the results to arrive at a comprehensive solution.

A number line serves as a powerful visual tool for representing solution sets, especially when dealing with inequalities and intervals. To illustrate the solution set x≥-5 on a number line, we follow a specific convention that clearly conveys the inclusion or exclusion of endpoints.

First, we draw a horizontal line, which represents the real number line. We then mark the key point of interest, which in this case is -5. This point acts as the boundary of our solution set.

Since our solution set includes all x-values greater than or equal to -5, we use a closed circle (or a filled-in circle) at -5 on the number line. This closed circle signifies that -5 itself is included in the solution set. If the solution set had been strictly greater than -5 (x>-5), we would have used an open circle at -5 to indicate that -5 is not included.

Next, we shade the portion of the number line that corresponds to x-values greater than -5. This shaded region extends indefinitely to the right, indicating that all numbers to the right of -5 are part of the solution set. The shading visually represents the infinite extent of the solution interval.

To further emphasize the unbounded nature of the solution set in the positive direction, we draw an arrow at the right end of the shaded region. This arrow symbolizes that the solution set continues indefinitely towards positive infinity.

In contrast, the portion of the number line to the left of -5 remains unshaded, indicating that these x-values are not part of the solution set. This clear demarcation visually separates the solutions from the non-solutions.

The number line representation provides an intuitive understanding of the solution set x≥-5. It visually confirms that the solution set is a continuous interval that includes -5 and all numbers greater than -5, extending indefinitely towards positive infinity. This visual aid is invaluable in comprehending the nature of solutions to inequalities and in communicating these solutions effectively.

In summary, the equation |x+5|=x+5 exemplifies the intricacies of absolute value equations and the importance of a systematic approach to solving them. By meticulously considering both cases, where x+5 is non-negative and negative, we have demonstrated that the solution set is x≥-5. This solution set encompasses all real numbers greater than or equal to -5, a fact that is vividly illustrated on the number line.

This exploration underscores the fundamental principles of absolute values, the significance of considering all possible scenarios, and the power of visual representations in mathematics. The process of solving this equation not only provides a concrete solution but also reinforces our understanding of mathematical concepts and problem-solving strategies. The journey from the equation to the solution set is a testament to the elegance and precision of mathematics.