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Answer :
- Set up the equation $4|x-5|+3 = 15$.
- Isolate the absolute value: $|x-5| = 3$.
- Split into two cases: $x-5 = 3$ and $x-5 = -3$.
- Solve for $x$ in each case, obtaining $x=8$ and $x=2$. The final answer is $\boxed{x=2, x=8}$.
### Explanation
1. Problem Analysis
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. Splitting into Cases
To solve the absolute value equation $|x-5|=3$, we consider two cases:
Case 1: $x-5 = 3$
Case 2: $x-5 = -3$
6. Solving for x in Each Case
Solving for $x$ in Case 1:
$$x-5 = 3$$$$x = 3 + 5$$$$x = 8$$
Solving for $x$ in Case 2:
$$x-5 = -3$$$$x = -3 + 5$$$$x = 2$$
7. 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 designed to produce parts that are 5 cm long, but a tolerance of 0.3 cm is allowed, the actual length $x$ of the part must satisfy the equation $|x-5| \le 0.3$. Solving this inequality tells us the acceptable range of lengths for the parts.
- Isolate the absolute value: $|x-5| = 3$.
- Split into two cases: $x-5 = 3$ and $x-5 = -3$.
- Solve for $x$ in each case, obtaining $x=8$ and $x=2$. The final answer is $\boxed{x=2, x=8}$.
### Explanation
1. Problem Analysis
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. Splitting into Cases
To solve the absolute value equation $|x-5|=3$, we consider two cases:
Case 1: $x-5 = 3$
Case 2: $x-5 = -3$
6. Solving for x in Each Case
Solving for $x$ in Case 1:
$$x-5 = 3$$$$x = 3 + 5$$$$x = 8$$
Solving for $x$ in Case 2:
$$x-5 = -3$$$$x = -3 + 5$$$$x = 2$$
7. 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 designed to produce parts that are 5 cm long, but a tolerance of 0.3 cm is allowed, the actual length $x$ of the part must satisfy the equation $|x-5| \le 0.3$. Solving this inequality tells us the acceptable range of lengths for the parts.
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