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Answer :
- Set up the equation $4|x-5| + 3 = 15$.
- Isolate the absolute value term: $4|x-5| = 12$, then $|x-5| = 3$.
- 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 $f(x) = 15$, which gives us the equation $4|x-5| + 3 = 15$.
3. Isolating the Absolute Value
Next, we isolate the absolute value term. Subtracting 3 from both sides of the equation gives us $4|x-5| = 15 - 3$, which simplifies to $4|x-5| = 12$.
4. Simplifying the Equation
Now, we divide both sides of the equation by 4 to get $|x-5| = \frac{12}{4}$, which simplifies to $|x-5| = 3$.
5. Solving for x
To solve the absolute value equation $|x-5| = 3$, we consider two cases:
Case 1: $x-5 = 3$. Adding 5 to both sides gives $x = 3 + 5 = 8$.
Case 2: $x-5 = -3$. Adding 5 to both sides gives $x = -3 + 5 = 2$.
6. Final Answer
Therefore, the values of $x$ for which $f(x) = 15$ are $x = 2$ and $x = 8$.
### Examples
Absolute value equations are useful in many real-world scenarios, such as determining tolerances in manufacturing. For example, if a machine is supposed to cut a metal rod to a length of 5 cm, but there is a tolerance of 3 cm, the actual length $x$ of the rod must satisfy $|x-5|
leq 3$. This means the length can be between 2 cm and 8 cm. Understanding absolute value equations helps engineers ensure that products meet the required specifications.
- Isolate the absolute value term: $4|x-5| = 12$, then $|x-5| = 3$.
- 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 $f(x) = 15$, which gives us the equation $4|x-5| + 3 = 15$.
3. Isolating the Absolute Value
Next, we isolate the absolute value term. Subtracting 3 from both sides of the equation gives us $4|x-5| = 15 - 3$, which simplifies to $4|x-5| = 12$.
4. Simplifying the Equation
Now, we divide both sides of the equation by 4 to get $|x-5| = \frac{12}{4}$, which simplifies to $|x-5| = 3$.
5. Solving for x
To solve the absolute value equation $|x-5| = 3$, we consider two cases:
Case 1: $x-5 = 3$. Adding 5 to both sides gives $x = 3 + 5 = 8$.
Case 2: $x-5 = -3$. Adding 5 to both sides gives $x = -3 + 5 = 2$.
6. Final Answer
Therefore, the values of $x$ for which $f(x) = 15$ are $x = 2$ and $x = 8$.
### Examples
Absolute value equations are useful in many real-world scenarios, such as determining tolerances in manufacturing. For example, if a machine is supposed to cut a metal rod to a length of 5 cm, but there is a tolerance of 3 cm, the actual length $x$ of the rod must satisfy $|x-5|
leq 3$. This means the length can be between 2 cm and 8 cm. Understanding absolute value equations helps engineers ensure that products meet the required specifications.
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