Solving The Absolute Value Equation 4|0.5x - 2.5| = 0
When we encounter an equation involving absolute values, like the one we're about to tackle, it's important to remember what the absolute value actually represents. The absolute value of a number is its distance from zero on the number line, and because distance can never be negative, the result of an absolute value operation is always non-negative. In the equation $4|0.5x - 2.5|=0$, we have an expression inside an absolute value. Our primary goal is to isolate this absolute value expression and then use the properties of absolute values to find the possible values for 'x'. Let's break down the steps involved in solving this specific equation. We'll start by simplifying the equation to make it easier to work with. The first step in solving any equation is to get the variable by itself. In this case, we have a coefficient of 4 multiplying the absolute value term. To isolate the absolute value, we'll divide both sides of the equation by 4. This is a fundamental algebraic manipulation that helps us peel away layers of the equation until we reach the core part we need to solve. Remember, whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain equality. So, dividing both sides of $4|0.5x - 2.5|=0$ by 4 gives us $|0.5x - 2.5| = 0/4$. Simplifying the right side, $0/4$ is simply 0. This leaves us with $|0.5x - 2.5| = 0$. Now we have the absolute value expression equal to zero. This is a crucial point. For the absolute value of an expression to be zero, the expression itself must be zero. There's no other way for a distance from zero to be zero. Think about it: if a number is not zero, its distance from zero is some positive value. Therefore, the only scenario where $|something| = 0$ is true is when $something = 0$. Applying this to our equation, we must have $0.5x - 2.5 = 0$. This simplifies our problem significantly, as we no longer need to consider two separate cases (like we would if the absolute value was equal to a positive number). Now, we're left with a simple linear equation to solve for 'x'. This is where basic algebra comes into play. To isolate 'x', we first want to get the term with 'x' by itself. We can do this by adding 2.5 to both sides of the equation: $0.5x - 2.5 + 2.5 = 0 + 2.5$. This simplifies to $0.5x = 2.5$. The final step to solve for 'x' is to eliminate the coefficient of 0.5. We can achieve this by dividing both sides of the equation by 0.5: $ rac{0.5x}{0.5} = rac{2.5}{0.5}$. Performing the division, we find that $x = 5$. So, the only solution to the equation $4|0.5x - 2.5|=0$ is $x=5$. It's always a good practice to check your solution by plugging it back into the original equation to ensure it holds true. Let's do that: $4|0.5(5) - 2.5| = 4|2.5 - 2.5| = 4|0| = 4(0) = 0$. The equation holds true, confirming that $x=5$ is indeed the correct solution. This methodical approach, starting with understanding the properties of absolute values and then applying basic algebraic steps, allows us to confidently solve equations of this nature.
Understanding Absolute Value Properties
The core of solving equations like $4|0.5x - 2.5|=0$ lies in understanding the fundamental properties of absolute value. Recall that the absolute value of a number, denoted as $|a|$, represents its distance from zero on the number line. This means $|a| less 0$; the absolute value is always non-negative. This non-negative property is critical. When we are presented with an equation of the form $|expression| = k$, we need to consider the value of 'k'. If 'k' is positive, say $|expression| = 5$, then 'expression' could be either 5 or -5, because $|5|=5$ and $|-5|=5$. This is why absolute value equations often yield two potential solutions. However, in our specific case, $|0.5x - 2.5| = 0$. This scenario is unique. For the absolute value of an expression to be equal to zero, the expression inside the absolute value bars must be equal to zero. There is no other possibility. If $0.5x - 2.5$ were any other number (positive or negative), its absolute value would be a positive number, not zero. This significantly simplifies the problem because it means we only need to solve one linear equation, not two. The steps we took were to first isolate the absolute value term by dividing both sides by 4, resulting in $|0.5x - 2.5| = 0$. Then, we set the expression inside the absolute value equal to zero: $0.5x - 2.5 = 0$. This is a linear equation, which we can solve using standard algebraic techniques. First, we add 2.5 to both sides to get $0.5x = 2.5$. Then, we divide both sides by 0.5 to find $x = 5$. The fact that the absolute value is equal to zero eliminates the need to consider multiple cases, making the solution process more direct. This illustrates how understanding the constraints imposed by the absolute value function is key to efficient problem-solving. The mathematical property that $|a|=0 ext{ if and only if } a=0$ is the cornerstone of solving this particular equation.
Step-by-Step Solution Breakdown
Let's meticulously walk through the process of solving the equation $4|0.5x - 2.5|=0$, ensuring clarity at each stage. Our objective is to find the value(s) of 'x' that satisfy this equation. The first crucial step is to isolate the absolute value expression. Currently, the absolute value term $|0.5x - 2.5|$ is being multiplied by 4. To undo this multiplication, we perform the inverse operation: division. We divide both sides of the equation by 4:
rac{4|0.5x - 2.5|}{4} = rac{0}{4}
This simplifies to:
Now we have the absolute value expression on one side and 0 on the other. This is where the specific property of the absolute value comes into play: the absolute value of an expression is zero if and only if the expression itself is zero. In other words, for $|0.5x - 2.5|$ to equal 0, the content within the absolute value bars, $0.5x - 2.5$, must be equal to 0.
So, we set up a simple linear equation:
The next step is to solve this linear equation for x. Our goal is to get 'x' by itself. First, we want to isolate the term containing 'x'. We can do this by adding 2.5 to both sides of the equation:
This simplifies to:
Finally, to find the value of 'x', we need to eliminate the coefficient of x, which is 0.5. We do this by dividing both sides of the equation by 0.5:
rac{0.5x}{0.5} = rac{2.5}{0.5}
Performing the division, we get:
Thus, the unique solution to the equation $4|0.5x - 2.5|=0$ is $x=5$. It's always a good habit to verify your solution by substituting it back into the original equation. Let's do that:
First, calculate the expression inside the absolute value:
Now, substitute this back into the equation:
Since the absolute value of 0 is 0:
The left side of the equation equals the right side (0 = 0), so our solution is correct. The step-by-step process, from isolating the absolute value to solving the resulting linear equation and verifying the answer, confirms that $x=5$ is the only solution.
Analyzing the Answer Choices
Now that we've solved the equation $4|0.5x - 2.5|=0$ and found that the unique solution is $x=5$, let's look at the provided answer choices to identify the correct one. The options are:
A. $x=1.25$ B. $x=-1.25 ext { or } x=1.25$ C. $x=-5 ext { or } x=5$ D. $x=5$
Our calculated solution is $x=5$. We need to find the option that matches this result. Option A suggests $x=1.25$. If we substitute this back into the original equation, we would get $4|0.5(1.25) - 2.5| = 4|0.625 - 2.5| = 4|-1.875| = 4(1.875) = 7.5$, which is not equal to 0. So, A is incorrect.
Option B suggests $x=-1.25 ext { or } x=1.25$. We've already seen that $x=1.25$ doesn't work. Let's test $x=-1.25$: $4|0.5(-1.25) - 2.5| = 4|-0.625 - 2.5| = 4|-3.125| = 4(3.125) = 12.5$, which is also not 0. So, B is incorrect.
Option C suggests $x=-5 ext { or } x=5$. We know that $x=5$ is a solution. Let's test $x=-5$: $4|0.5(-5) - 2.5| = 4|-2.5 - 2.5| = 4|-5| = 4(5) = 20$, which is not 0. So, C is incorrect because it includes an extraneous solution.
Option D suggests $x=5$. This perfectly matches our derived solution. As we verified earlier, substituting $x=5$ into the original equation yields $4|0.5(5) - 2.5| = 4|2.5 - 2.5| = 4|0| = 0$. This confirms that $x=5$ is the correct and only solution.
The key takeaway here is that when an absolute value equation simplifies to $|expression| = 0$, there is only one possibility: $expression = 0$. This eliminates the need to consider positive and negative cases, which are typical for equations where $|expression| = ext{positive number}$. The presence of choices like C, which include both a correct and an incorrect solution, highlights the importance of carefully checking all potential answers derived from the initial algebraic steps.
Conclusion
In conclusion, solving the absolute value equation $4|0.5x - 2.5|=0$ involves a straightforward application of algebraic principles and an understanding of absolute value properties. The equation simplifies by first isolating the absolute value term, leading to $|0.5x - 2.5|=0$. Crucially, for an absolute value to be zero, the expression within it must also be zero. This directed us to solve the linear equation $0.5x - 2.5 = 0$. Through a series of simple steps β adding 2.5 to both sides and then dividing by 0.5 β we arrived at the unique solution $x=5$. Verifying this solution by plugging it back into the original equation confirmed its accuracy. Among the given options, only D. $x=5$ correctly represents this solution. Understanding that $|a|=0 ext{ implies } a=0$ is fundamental to solving this type of equation efficiently and avoiding unnecessary complexity. For further exploration into algebraic equations and their solutions, you can visit Khan Academy for comprehensive resources and practice exercises.
