Answer :
- Isolate the absolute value term: $4|x+5| = 16$.
- Divide by 4: $|x+5| = 4$.
- Solve for $x$ when $x+5 = 4$, which gives $x = -1$.
- Solve for $x$ when $x+5 = -4$, which gives $x = -9$. The final answer is $\boxed{x=-1 \text{ and } x=-9}$
### Explanation
1. Understanding the Problem
We are given the equation $4|x+5|+8=24$. Our goal is 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:$$4|x+5| = 24 - 8$$$$4|x+5| = 16$$
3. Simplifying the Equation
Next, divide both sides by 4 to further isolate the absolute value:$$|x+5| = \frac{16}{4}$$$$|x+5| = 4$$
4. Solving for x in Both Cases
Now we consider the two cases for the absolute value. Case 1: The expression inside the absolute value is positive or zero, i.e., $x+5 = 4$. Solving for $x$, we subtract 5 from both sides:$$x = 4 - 5$$$$x = -1$$ Case 2: The expression inside the absolute value is negative, i.e., $x+5 = -4$. Solving for $x$, we subtract 5 from both sides:$$x = -4 - 5$$$$x = -9$$
5. Final Answer
Therefore, the solutions are $x = -1$ and $x = -9$. Comparing these solutions with the given options, we see that the correct answer is D.
### Examples
Absolute value equations are useful in many real-world scenarios, such as calculating tolerances in engineering or determining distances that can vary in either direction from a fixed point. For example, if you are manufacturing a part that needs to be 5 cm long with a tolerance of 0.1 cm, the actual length $x$ must satisfy $|x - 5| <= 0.1$. Solving this inequality helps determine the acceptable range of lengths for the part.
- Divide by 4: $|x+5| = 4$.
- Solve for $x$ when $x+5 = 4$, which gives $x = -1$.
- Solve for $x$ when $x+5 = -4$, which gives $x = -9$. The final answer is $\boxed{x=-1 \text{ and } x=-9}$
### Explanation
1. Understanding the Problem
We are given the equation $4|x+5|+8=24$. Our goal is 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:$$4|x+5| = 24 - 8$$$$4|x+5| = 16$$
3. Simplifying the Equation
Next, divide both sides by 4 to further isolate the absolute value:$$|x+5| = \frac{16}{4}$$$$|x+5| = 4$$
4. Solving for x in Both Cases
Now we consider the two cases for the absolute value. Case 1: The expression inside the absolute value is positive or zero, i.e., $x+5 = 4$. Solving for $x$, we subtract 5 from both sides:$$x = 4 - 5$$$$x = -1$$ Case 2: The expression inside the absolute value is negative, i.e., $x+5 = -4$. Solving for $x$, we subtract 5 from both sides:$$x = -4 - 5$$$$x = -9$$
5. Final Answer
Therefore, the solutions are $x = -1$ and $x = -9$. Comparing these solutions with the given options, we see that the correct answer is D.
### Examples
Absolute value equations are useful in many real-world scenarios, such as calculating tolerances in engineering or determining distances that can vary in either direction from a fixed point. For example, if you are manufacturing a part that needs to be 5 cm long with a tolerance of 0.1 cm, the actual length $x$ must satisfy $|x - 5| <= 0.1$. Solving this inequality helps determine the acceptable range of lengths for the part.