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