Answer :
- Solve $4|x-5|+3 = 15$ by isolating the absolute value and splitting into two cases: $x-5 = 3$ and $x-5 = -3$.
- Find the solutions for the first equation: $x = 2$ and $x = 8$.
- Solve $-0.5|2x+2|+1 = 6$ by isolating the absolute value and obtaining $|2x+2| = -10$.
- Since the absolute value cannot be negative, there is no solution for the second equation.
### Explanation
1. Problem Analysis
We are given two functions and asked to find the values of $x$ for which $f(x) = 15$ for the first function and $f(x) = 6$ for the second function. We will solve each equation separately.
2. Solving the First Equation
First, let's solve $4|x-5|+3 = 15$ for $x$.
Subtract 3 from both sides: $4|x-5| = 15 - 3$, which simplifies to $4|x-5| = 12$.
Divide both sides by 4: $|x-5| = \frac{12}{4}$, which simplifies to $|x-5| = 3$.
Now, we consider two cases:
Case 1: $x-5 = 3$. Adding 5 to both sides gives $x = 3 + 5 = 8$.
Case 2: $x-5 = -3$. Adding 5 to both sides gives $x = -3 + 5 = 2$.
So, the solutions for the first equation are $x = 2$ and $x = 8$.
3. Solving the Second Equation
Next, let's solve $-0.5|2x+2|+1 = 6$ for $x$.
Subtract 1 from both sides: $-0.5|2x+2| = 6 - 1$, which simplifies to $-0.5|2x+2| = 5$.
Divide both sides by -0.5: $|2x+2| = \frac{5}{-0.5}$, which simplifies to $|2x+2| = -10$.
Since the absolute value of any expression must be non-negative, there is no solution to the equation $|2x+2| = -10$.
4. Final Answer
Therefore, the values of $x$ for which $f(x) = 15$ are $x=2$ and $x=8$, and there is no solution for the second equation $f(x) = 6$.
### Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or tolerances in engineering and manufacturing. For example, if you are designing a component that needs to fit within a certain tolerance, you can use absolute value equations to determine the acceptable range of measurements. Similarly, in navigation, absolute value can be used to calculate the distance between two points, regardless of direction.
- Find the solutions for the first equation: $x = 2$ and $x = 8$.
- Solve $-0.5|2x+2|+1 = 6$ by isolating the absolute value and obtaining $|2x+2| = -10$.
- Since the absolute value cannot be negative, there is no solution for the second equation.
### Explanation
1. Problem Analysis
We are given two functions and asked to find the values of $x$ for which $f(x) = 15$ for the first function and $f(x) = 6$ for the second function. We will solve each equation separately.
2. Solving the First Equation
First, let's solve $4|x-5|+3 = 15$ for $x$.
Subtract 3 from both sides: $4|x-5| = 15 - 3$, which simplifies to $4|x-5| = 12$.
Divide both sides by 4: $|x-5| = \frac{12}{4}$, which simplifies to $|x-5| = 3$.
Now, we consider two cases:
Case 1: $x-5 = 3$. Adding 5 to both sides gives $x = 3 + 5 = 8$.
Case 2: $x-5 = -3$. Adding 5 to both sides gives $x = -3 + 5 = 2$.
So, the solutions for the first equation are $x = 2$ and $x = 8$.
3. Solving the Second Equation
Next, let's solve $-0.5|2x+2|+1 = 6$ for $x$.
Subtract 1 from both sides: $-0.5|2x+2| = 6 - 1$, which simplifies to $-0.5|2x+2| = 5$.
Divide both sides by -0.5: $|2x+2| = \frac{5}{-0.5}$, which simplifies to $|2x+2| = -10$.
Since the absolute value of any expression must be non-negative, there is no solution to the equation $|2x+2| = -10$.
4. Final Answer
Therefore, the values of $x$ for which $f(x) = 15$ are $x=2$ and $x=8$, and there is no solution for the second equation $f(x) = 6$.
### Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or tolerances in engineering and manufacturing. For example, if you are designing a component that needs to fit within a certain tolerance, you can use absolute value equations to determine the acceptable range of measurements. Similarly, in navigation, absolute value can be used to calculate the distance between two points, regardless of direction.