Absolute Value Equation: Finding The Second Solution

Manuela has taken a great first step in solving the absolute value equation $3-2|0.5 x+1.5|=2$. She has correctly isolated the absolute value expression and found one of the possible solutions, $x = -2$. However, absolute value equations often have two solutions because the expression inside the absolute value can be equal to either a positive or a negative value. Let's dive into how we can find that other crucial solution and understand why it exists.

Understanding Absolute Value Equations

Before we jump into finding Manuela's second solution, it's essential to grasp what an absolute value actually represents. The absolute value of a number, denoted by vertical bars $| \cdot |$, tells us its distance from zero on the number line. For instance, $|5| = 5$ and $|-5| = 5$. This is why an absolute value equation like $|\text{expression}| = c$ (where $c$ is a positive number) can have two solutions: the expression inside the bars can equal $c$, or it can equal $-c$. In Manuela's case, her equation simplifies to $|0.5 x+1.5| = 0.5$. This means that the quantity $0.5x + 1.5$ must be a number whose distance from zero is $0.5$. On a number line, there are two such numbers: $0.5$ and $-0.5$. Manuela's work correctly followed the path for $0.5x + 1.5 = 0.5$, leading to her solution $x = -2$. Now, we need to explore the other possibility.

Finding the Second Solution

Manuela's last step in her shown work was setting the expression inside the absolute value equal to $0.5$: $0.5 x+1.5 = 0.5$. This is a perfectly valid step, and it led her to one correct solution. To find the other solution, we must consider the case where the expression inside the absolute value is equal to the negative of $0.5$. So, we set up our second equation:

$$0.5 x+1.5 = -0.5$$

Now, we solve this equation for $x$, just as Manuela did with her first equation. First, subtract $1.5$ from both sides:

$$0.5 x = -0.5 - 1.5$$

$$0.5 x = -2$$

Next, divide both sides by $0.5$ (which is the same as multiplying by 2):

$$x = \frac{-2}{0.5}$$

$$x = -4$$

So, the other solution to the equation $3-2|0.5 x+1.5|=2$ is $x = -4$. It's always a good idea to check both solutions in the original equation to ensure they are correct. Let's do that!

Checking the Solutions

Checking Manuela's solution ($x = -2$):

$$3-2|0.5(-2)+1.5| = 3-2|-1+1.5| = 3-2|0.5| = 3-2(0.5) = 3-1 = 2$$

This confirms that $x = -2$ is indeed a correct solution.

Checking our new solution ($x = -4$):

$$3-2|0.5(-4)+1.5| = 3-2|-2+1.5| = 3-2|-0.5| = 3-2(0.5) = 3-1 = 2$$

This also confirms that $x = -4$ is a correct solution. Both values satisfy the original equation.

Why Two Solutions? The Power of Absolute Value

The concept of absolute value is fundamental here. When we see an equation like $|A| = B$ (where $B > 0$), we are essentially saying that the quantity $A$ is $B$ units away from zero. On a number line, there are always two positions that are $B$ units away from zero: one to the right (positive $B$) and one to the left (negative $B$). Therefore, $A$ can be equal to $B$, or $A$ can be equal to $-B$. This is why most absolute value equations that simplify to $|\text{expression}| = \text{positive number}$ will yield two distinct solutions. In Manuela's problem, the expression $0.5x + 1.5$ represents the quantity that, when its absolute value is taken, equals $0.5$. Since $0.5x + 1.5$ could be $0.5$ OR $-0.5$, we get two distinct values for $x$. It's a common pitfall for students to only solve for the positive case and forget the negative case, thus missing out on a perfectly valid solution. Remembering this duality is key to mastering absolute value equations.

Common Mistakes and How to Avoid Them

One of the most frequent errors when solving absolute value equations is forgetting to set up the second equation with the negative value. After isolating the absolute value expression, say $|expression| = c$, students might only solve $expression = c$ and stop. This leads to missing one of the two possible solutions. Always remember the definition of absolute value: it's about distance, and distance can be measured in two directions from zero. Another mistake can occur during the algebraic manipulation. Errors in signs or arithmetic when solving either the positive or negative case can lead to incorrect solutions. Double-checking your algebra and, most importantly, plugging your final answers back into the original equation is the best way to catch these errors. If a solution doesn't work in the original equation, it's either an extraneous solution (which can happen in some types of absolute value problems, though not typically in this simpler form) or you made an algebraic mistake somewhere along the line.

Conclusion: The Complete Picture

Manuela's journey to solve $3-2|0.5 x+1.5|=2$ beautifully illustrates a core concept in algebra: the dual nature of absolute value. By correctly isolating the absolute value term and then considering both the positive and negative possibilities for the expression within, we arrive at the complete set of solutions. Her initial work yielded $x = -2$, and by exploring the alternative where $0.5x + 1.5 = -0.5$, we discovered the second solution, $x = -4$. Understanding this principle is not just about solving one problem; it's about building a solid foundation for more complex mathematical concepts. Always remember that $|A| = c$ (for $c > 0$) implies $A = c$ or $A = -c$. This simple rule unlocks the full set of solutions for absolute value equations.

For further exploration into solving equations and algebraic concepts, I recommend visiting resources like Khan Academy, which offers comprehensive lessons and practice exercises on a wide range of mathematical topics. You can find excellent materials on absolute value equations and beyond at Khan Academy.