Answer :
- Isolate the absolute value term: $|x+5| = 13$.
- Consider two cases: $x+5 = 13$ and $x+5 = -13$.
- Solve for $x$ in each case: $x = 8$ and $x = -18$.
- The solutions are $x = 8$ and $x = -18$, so the answer is $\boxed{x=8 \text{ and } x=-18}$.
### Explanation
1. Understanding the problem
We are given the equation $|x+5|-6=7$ and we need to find the values of $x$ that satisfy this equation.
2. Isolating the absolute value
First, we isolate the absolute value term by adding 6 to both sides of the equation:$$|x+5| - 6 + 6 = 7 + 6$$$$|x+5| = 13$$
3. Solving for x in both cases
Now, we consider two cases.
Case 1: $x+5$ is positive or zero, so $|x+5| = x+5$. Then we have:
$$x+5 = 13$$
Subtracting 5 from both sides gives:
$$x = 13 - 5$$$$x = 8$$
Case 2: $x+5$ is negative, so $|x+5| = -(x+5)$. Then we have:
$$-(x+5) = 13$$
Multiplying both sides by -1 gives:
$$x+5 = -13$$
Subtracting 5 from both sides gives:
$$x = -13 - 5$$$$x = -18$$
4. Final Answer
Therefore, the solutions are $x=8$ and $x=-18$.
### Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or errors. For example, if you're manufacturing parts and need them to be within a certain tolerance, you can use absolute value to express the acceptable range of measurements. Similarly, in physics, absolute value can be used to calculate the magnitude of a vector, regardless of its direction.
- Consider two cases: $x+5 = 13$ and $x+5 = -13$.
- Solve for $x$ in each case: $x = 8$ and $x = -18$.
- The solutions are $x = 8$ and $x = -18$, so the answer is $\boxed{x=8 \text{ and } x=-18}$.
### Explanation
1. Understanding the problem
We are given the equation $|x+5|-6=7$ and we need to find the values of $x$ that satisfy this equation.
2. Isolating the absolute value
First, we isolate the absolute value term by adding 6 to both sides of the equation:$$|x+5| - 6 + 6 = 7 + 6$$$$|x+5| = 13$$
3. Solving for x in both cases
Now, we consider two cases.
Case 1: $x+5$ is positive or zero, so $|x+5| = x+5$. Then we have:
$$x+5 = 13$$
Subtracting 5 from both sides gives:
$$x = 13 - 5$$$$x = 8$$
Case 2: $x+5$ is negative, so $|x+5| = -(x+5)$. Then we have:
$$-(x+5) = 13$$
Multiplying both sides by -1 gives:
$$x+5 = -13$$
Subtracting 5 from both sides gives:
$$x = -13 - 5$$$$x = -18$$
4. Final Answer
Therefore, the solutions are $x=8$ and $x=-18$.
### Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or errors. For example, if you're manufacturing parts and need them to be within a certain tolerance, you can use absolute value to express the acceptable range of measurements. Similarly, in physics, absolute value can be used to calculate the magnitude of a vector, regardless of its direction.