Answer :
- Set up the equation $4|x-5|+3 = 15$.
- Isolate the absolute value term: $|x-5| = 3$.
- Solve for 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. Problem Setup
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. Considering Two 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 Case 1
For Case 1, we solve for $x$:
$$x-5 = 3$$$$x = 3 + 5$$$$x = 8$$
7. Solving Case 2
For Case 2, we solve for $x$:
$$x-5 = -3$$$$x = -3 + 5$$$$x = 2$$
8. 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 various real-world scenarios, such as determining the tolerance or acceptable range of values in manufacturing or engineering. For example, if a machine part needs to be within 0.05 cm of a specific length, the absolute value equation can help determine the acceptable lengths of the manufactured parts. Another example is in navigation, where absolute values can be used to calculate distances regardless of direction.
- Isolate the absolute value term: $|x-5| = 3$.
- Solve for 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. Problem Setup
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. Considering Two 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 Case 1
For Case 1, we solve for $x$:
$$x-5 = 3$$$$x = 3 + 5$$$$x = 8$$
7. Solving Case 2
For Case 2, we solve for $x$:
$$x-5 = -3$$$$x = -3 + 5$$$$x = 2$$
8. 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 various real-world scenarios, such as determining the tolerance or acceptable range of values in manufacturing or engineering. For example, if a machine part needs to be within 0.05 cm of a specific length, the absolute value equation can help determine the acceptable lengths of the manufactured parts. Another example is in navigation, where absolute values can be used to calculate distances regardless of direction.