Answer :
- Set up the equation $4|x-5| + 3 = 15$.
- Isolate the absolute value: $|x-5| = 3$.
- Solve for $x$ by considering two cases: $x-5 = 3$ and $x-5 = -3$.
- The solutions are $x = 2$ and $x = 8$, so the final answer is $\boxed{x=2, x=8}$.
### Explanation
1. Understanding the Problem
We are given the function $f(x) = 4|x-5| + 3$ and we want to find the values of $x$ for which $f(x) = 15$. This involves solving an absolute value equation.
2. Setting up the Equation
First, we set $f(x)$ equal to 15: $$4|x-5| + 3 = 15$$
3. Isolating the Absolute Value
Next, we isolate the absolute value term. Subtract 3 from both sides of the equation:$$4|x-5| = 15 - 3$$$$4|x-5| = 12$$
4. Simplifying the Equation
Now, divide both sides by 4:$$|x-5| = \frac{12}{4}$$$$|x-5| = 3$$
5. Solving for x
To solve the absolute value equation $|x-5| = 3$, we consider two cases:
Case 1: $x-5 = 3$
Add 5 to both sides:$$x = 3 + 5$$$$x = 8$$
Case 2: $x-5 = -3$
Add 5 to both sides:$$x = -3 + 5$$$$x = 2$$
6. Final Answer
Therefore, the values of $x$ for which $f(x) = 15$ are $x = 2$ and $x = 8$.
### Examples
Absolute value functions are used in various real-world scenarios, such as calculating distances or deviations from a target value. For example, in manufacturing, if a machine is set to produce parts of a certain length, the absolute value function can be used to determine how much each part deviates from the target length, regardless of whether the part is too long or too short. This helps in quality control and ensuring that parts meet the required specifications.
- Isolate the absolute value: $|x-5| = 3$.
- Solve for $x$ by considering two cases: $x-5 = 3$ and $x-5 = -3$.
- The solutions are $x = 2$ and $x = 8$, so the final answer is $\boxed{x=2, x=8}$.
### Explanation
1. Understanding the Problem
We are given the function $f(x) = 4|x-5| + 3$ and we want to find the values of $x$ for which $f(x) = 15$. This involves solving an absolute value equation.
2. Setting up the Equation
First, we set $f(x)$ equal to 15: $$4|x-5| + 3 = 15$$
3. Isolating the Absolute Value
Next, we isolate the absolute value term. Subtract 3 from both sides of the equation:$$4|x-5| = 15 - 3$$$$4|x-5| = 12$$
4. Simplifying the Equation
Now, divide both sides by 4:$$|x-5| = \frac{12}{4}$$$$|x-5| = 3$$
5. Solving for x
To solve the absolute value equation $|x-5| = 3$, we consider two cases:
Case 1: $x-5 = 3$
Add 5 to both sides:$$x = 3 + 5$$$$x = 8$$
Case 2: $x-5 = -3$
Add 5 to both sides:$$x = -3 + 5$$$$x = 2$$
6. Final Answer
Therefore, the values of $x$ for which $f(x) = 15$ are $x = 2$ and $x = 8$.
### Examples
Absolute value functions are used in various real-world scenarios, such as calculating distances or deviations from a target value. For example, in manufacturing, if a machine is set to produce parts of a certain length, the absolute value function can be used to determine how much each part deviates from the target length, regardless of whether the part is too long or too short. This helps in quality control and ensuring that parts meet the required specifications.