Solving Absolute Value Equations: Which Statement Is True?

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Hey everyone! Let's dive into the world of absolute value equations and figure out which statement is actually true. This can seem tricky, but we'll break it down step by step. We'll explore each equation, simplify them, and then determine the number of solutions. Stick with me, and you'll be solving these like a pro in no time!

A. The equation $-3|2x + 1.2| = -1$ has no solution.

Let's get started by examining the first statement: The equation $-3|2x + 1.2| = -1$ has no solution. To determine if this is true, we first need to isolate the absolute value expression. Guys, remember that the absolute value of anything is always non-negative (that means zero or positive). So, we're looking for cases where this holds true, or where it doesn't.

To isolate the absolute value, we'll divide both sides of the equation by -3. This gives us: $|2x + 1.2| = \frac{-1}{-3}$, which simplifies to $|2x + 1.2| = \frac{1}{3}$. Now, we have the absolute value expression by itself. To solve this, we need to consider two possibilities:

  • The expression inside the absolute value is equal to 13\frac{1}{3}: $2x + 1.2 = \frac{1}{3}$
  • The expression inside the absolute value is equal to −13-\frac{1}{3}: $2x + 1.2 = -\frac{1}{3}$

Let's solve the first case: $2x + 1.2 = \frac1}{3}$. First, convert 1.2 to a fraction, which is 65\frac{6}{5}. Then, we have $2x + \frac{6}{5} = \frac{1}{3}$. Subtract 65\frac{6}{5} from both sides $2x = \frac{13} - \frac{6}{5}$. To subtract the fractions, we need a common denominator, which is 15. So, we get $2x = \frac{5}{15} - \frac{18}{15}$, which simplifies to $2x = -\frac{13}{15}$. Now, divide both sides by 2 $x = -\frac{13{30}$.

Now, let's solve the second case: $2x + 1.2 = -\frac1}{3}$. Again, convert 1.2 to 65\frac{6}{5}, so we have $2x + \frac{6}{5} = -\frac{1}{3}$. Subtract 65\frac{6}{5} from both sides $2x = -\frac{13} - \frac{6}{5}$. Using the common denominator of 15, we get $2x = -\frac{5}{15} - \frac{18}{15}$, which simplifies to $2x = -\frac{23}{15}$. Divide both sides by 2 $x = -\frac{23{30}$.

So, we have found two solutions for this equation: $x = -\frac{13}{30}$ and $x = -\frac{23}{30}$. This means that statement A, which claims the equation has no solution, is false. We've shown it has two solutions. This was a crucial step in determining the correct answer, and it highlights the importance of carefully working through each possibility when dealing with absolute value equations.

B. The equation $3.5|6x - 2| = 3.5$ has one solution.

Next up, let’s tackle statement B: The equation $3.5|6x - 2| = 3.5$ has one solution. To determine if this statement is true, we need to solve the equation and see how many solutions we find. The first step, as before, is to isolate the absolute value term. We can do this by dividing both sides of the equation by 3.5. This gives us: $|6x - 2| = \frac{3.5}{3.5}$, which simplifies to $|6x - 2| = 1$.

Now that we have the absolute value isolated, we need to consider two cases, just like before:

  • The expression inside the absolute value is equal to 1: $6x - 2 = 1$
  • The expression inside the absolute value is equal to -1: $6x - 2 = -1$

Let’s solve the first case: $6x - 2 = 1$. Add 2 to both sides: $6x = 3$. Divide both sides by 6: $x = \frac{3}{6}$, which simplifies to $x = \frac{1}{2}$.

Now, let's solve the second case: $6x - 2 = -1$. Add 2 to both sides: $6x = 1$. Divide both sides by 6: $x = \frac{1}{6}$.

So, we have found two solutions for this equation: $x = \frac{1}{2}$ and $x = \frac{1}{6}$. This means that statement B, which claims the equation has one solution, is false. Just like the previous equation, this one has two solutions, not one. Understanding this process is essential for solving absolute value equations, and it's important to avoid common mistakes like assuming there's only one solution. Remember to always consider both positive and negative cases!

C. The equation $5|-3.1x + 6.9| = -3.5$ has two solutions.

Alright, let's examine statement C: The equation $5|-3.1x + 6.9| = -3.5$ has two solutions. Following our established method, the first thing we need to do is isolate the absolute value term. We do this by dividing both sides of the equation by 5: $|-3.1x + 6.9| = \frac{-3.5}{5}$, which simplifies to $|-3.1x + 6.9| = -0.7$.

Now, guys, this is a crucial point! We have an absolute value expression equal to a negative number. Remember, the absolute value of any expression is always non-negative (zero or positive). It can never be a negative number. Therefore, there is no solution to this equation. Think about it – the distance from any number to zero can't be negative!

So, statement C, which claims the equation has two solutions, is definitely false. This case is a great reminder that you always need to check the result after isolating the absolute value. If you find the absolute value equal to a negative number, you can immediately conclude there are no solutions. This key insight can save you time and effort in solving these problems, and it's important to always keep this rule in mind. It's essential for understanding absolute value equations.

D. The equation $-0.3|3 + 8x| = -0.9$ has two solutions.

Finally, let's analyze statement D: The equation $-0.3|3 + 8x| = -0.9$ has two solutions. Just like before, our first step is to isolate the absolute value expression. To do this, we divide both sides of the equation by -0.3: $|3 + 8x| = \frac{-0.9}{-0.3}$, which simplifies to $|3 + 8x| = 3$.

Now that the absolute value is isolated, we need to consider two cases:

  • The expression inside the absolute value is equal to 3: $3 + 8x = 3$
  • The expression inside the absolute value is equal to -3: $3 + 8x = -3$

Let's solve the first case: $3 + 8x = 3$. Subtract 3 from both sides: $8x = 0$. Divide both sides by 8: $x = 0$.

Now, let's solve the second case: $3 + 8x = -3$. Subtract 3 from both sides: $8x = -6$. Divide both sides by 8: $x = \frac{-6}{8}$, which simplifies to $x = -\frac{3}{4}$.

We have found two solutions for this equation: $x = 0$ and $x = -\frac{3}{4}$. Therefore, statement D, which claims the equation has two solutions, is true! This is the correct answer. We've successfully worked through all the options and found the one that holds up. Guys, this demonstrates the power of breaking down a problem into smaller steps and carefully considering each possibility. It's essential for accuracy in math and in life!

Conclusion

After carefully analyzing each statement, we've determined that statement D is the only true statement. The equation $-0.3|3 + 8x| = -0.9$ indeed has two solutions. We tackled each equation by isolating the absolute value, considering both positive and negative cases, and solving for x. This process has given us a solid understanding of how to solve absolute value equations and how to avoid common pitfalls. Remember, when dealing with absolute values, always consider both possibilities and always check for extraneous solutions. Keep practicing, and you'll become a master of absolute value equations in no time! I hope this was helpful for you, and remember, math can be fun when you break it down into manageable steps!