Let​ a, b,​ c, and x be real numbers.
a. How is solving StartAbsoluteValue ax EndAbsoluteValue plus b equals c different from solving StartAbsoluteValue ax plus b EndAbsoluteValue equals c​?
b. How is solving StartAbsoluteValue ax EndAbsoluteValue plus b less than or equals c different from solving StartAbsoluteValue ax plus b EndAbsoluteValue greater than or equals c​?
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Part 1
a. How is solving StartAbsoluteValue ax EndAbsoluteValue plus b equals c different from solving StartAbsoluteValue ax plus b EndAbsoluteValue equals c​?
A.
When solving StartAbsoluteValue ax EndAbsoluteValue plus b equals c​, b has to be subtracted from both sides first to isolate the absolute value expression.
B.
Solving StartAbsoluteValue ax plus b EndAbsoluteValue equals c involves solving two equations that do not have absolute values. Solving StartAbsoluteValue ax EndAbsoluteValue plus b equals c involves solving only one such equation.
C.
Solving StartAbsoluteValue ax EndAbsoluteValue plus b equals c involves solving two equations that do not have absolute values. Solving StartAbsoluteValue ax plus b EndAbsoluteValue equals c involves solving only one such equation.
D.
When solving StartAbsoluteValue ax plus b EndAbsoluteValue equals c​, b has to be subtracted from both sides first to isolate the absolute value expression.

### Part 1a: Understanding the Differences

Let's break down the two equations to understand how they differ in terms of solving them:

1. **Equation 1**: \(|ax| + b = c\)
   - **Step**: The first step is to isolate the absolute value expression. To do this, you subtract \(b\) from both sides: 
     \[
     |ax| = c - b
     \]
   - **Next Step**: Now, you can proceed by splitting this into two cases, since \(|ax| = c - b\) implies:
     \[
     ax = c - b \quad \text{or} \quad ax = -(c - b)
     \]
   - Finally, solve for \(x\) in both cases.

2. **Equation 2**: \(|ax + b| = c\)
   - **Step**: Here, the absolute value expression is already isolated. Therefore, you can immediately split this into two cases:
     \[
     ax + b = c \quad \text{or} \quad ax + b = -c
     \]
   - **Next Step**: Solve each equation for \(x\) by isolating \(x\) in both cases.

### Choosing the Correct Answer:
- **Option A**: Correct. For \(|ax| + b = c\), you need to subtract \(b\) first to isolate the absolute value expression.
- **Option B**: Incorrect. Both forms involve solving two linear equations when you remove the absolute values.
- **Option C**: Incorrect. This is the reverse of the correct scenario.
- **Option D**: Incorrect. The description is correct for \(|ax + b| = c\), but the situation described doesn't fit.

**Correct Answer**: **A.**

### Follow-Up

Would you like further details or have any questions? Here are some related questions to deepen your understanding:

1. How do the solutions change if \(c\) is negative in \(|ax + b| = c\)?
2. What is the geometric interpretation of \(|ax| + b = c\) versus \(|ax + b| = c\)?
3. How does the solution process change if \(b > c\) in \(|ax| + b = c\)?
4. What happens if \(b = 0\) in both equations?
5. How do these approaches change if the equation involves inequalities instead of equalities?

**Tip**: Always isolate the absolute value expression first before

Let​ a, b,​ c, and x be real numbers.

a. How is solving |ax| + b = c different from solving |ax + b| = c​?

b. How is solving |ax| + b ≤ c different from solving |ax + b| ≥ c​?

Explore the differences between solving the absolute value equations |ax| + b = c and |ax + b| = c, as well as the inequalities |ax| + b ≤ c and |ax + b| ≥ c. This detailed guide covers algebraic concepts, key formulas, and theorems suitable for students in Grades 8-10.

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