Answer :
- Isolate the absolute value term: $4|x+7| = 24$.
- Divide by 4: $|x+7| = 6$.
- Solve for $x$ in both cases: $x+7 = 6$ and $-(x+7) = 6$.
- The solutions are $x = -1$ and $x = -13$, so the answer is $\boxed{x=-1 \text{ and } x=-13}$.
### Explanation
1. Understanding the Problem
We are given the equation $4|x+7|+8=32$ and asked to solve for $x$. This equation involves an absolute value, which means we need to consider two separate cases to find all possible solutions.
2. Isolating the Absolute Value
First, we isolate the absolute value term. Subtract 8 from both sides of the equation: $$4|x+7| + 8 - 8 = 32 - 8$$ $$4|x+7| = 24$$
3. Simplifying the Equation
Next, divide both sides of the equation by 4: $$\frac{4|x+7|}{4} = \frac{24}{4}$$ $$|x+7| = 6$$
4. Solving for x in Both Cases
Now, we consider the two cases for the absolute value:
Case 1: $x+7 \ge 0$, which means $|x+7| = x+7$. So, we have: $$x+7 = 6$$ Subtract 7 from both sides: $$x = 6 - 7$$ $$x = -1$$
Case 2: $x+7 < 0$, which means $|x+7| = -(x+7)$. So, we have: $$-(x+7) = 6$$ Multiply both sides by -1: $$x+7 = -6$$ Subtract 7 from both sides: $$x = -6 - 7$$ $$x = -13$$
5. Finding the Solution
Therefore, the solutions are $x = -1$ and $x = -13$. Comparing these solutions with the given options, we see that option C matches our solutions.
6. Final Answer
The solutions to the equation $4|x+7|+8=32$ are $x=-1$ and $x=-13$.
### Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or tolerances in engineering and manufacturing. For example, if you're designing a part that needs to be exactly 5 cm long, but you allow for a tolerance of 0.1 cm, you can use an absolute value equation to describe the acceptable range of lengths. The equation would be $|x - 5| <= 0.1$, where x is the actual length of the part. Solving this equation helps determine the minimum and maximum acceptable lengths.
- Divide by 4: $|x+7| = 6$.
- Solve for $x$ in both cases: $x+7 = 6$ and $-(x+7) = 6$.
- The solutions are $x = -1$ and $x = -13$, so the answer is $\boxed{x=-1 \text{ and } x=-13}$.
### Explanation
1. Understanding the Problem
We are given the equation $4|x+7|+8=32$ and asked to solve for $x$. This equation involves an absolute value, which means we need to consider two separate cases to find all possible solutions.
2. Isolating the Absolute Value
First, we isolate the absolute value term. Subtract 8 from both sides of the equation: $$4|x+7| + 8 - 8 = 32 - 8$$ $$4|x+7| = 24$$
3. Simplifying the Equation
Next, divide both sides of the equation by 4: $$\frac{4|x+7|}{4} = \frac{24}{4}$$ $$|x+7| = 6$$
4. Solving for x in Both Cases
Now, we consider the two cases for the absolute value:
Case 1: $x+7 \ge 0$, which means $|x+7| = x+7$. So, we have: $$x+7 = 6$$ Subtract 7 from both sides: $$x = 6 - 7$$ $$x = -1$$
Case 2: $x+7 < 0$, which means $|x+7| = -(x+7)$. So, we have: $$-(x+7) = 6$$ Multiply both sides by -1: $$x+7 = -6$$ Subtract 7 from both sides: $$x = -6 - 7$$ $$x = -13$$
5. Finding the Solution
Therefore, the solutions are $x = -1$ and $x = -13$. Comparing these solutions with the given options, we see that option C matches our solutions.
6. Final Answer
The solutions to the equation $4|x+7|+8=32$ are $x=-1$ and $x=-13$.
### Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or tolerances in engineering and manufacturing. For example, if you're designing a part that needs to be exactly 5 cm long, but you allow for a tolerance of 0.1 cm, you can use an absolute value equation to describe the acceptable range of lengths. The equation would be $|x - 5| <= 0.1$, where x is the actual length of the part. Solving this equation helps determine the minimum and maximum acceptable lengths.