Part 1

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

Correct Answer: A.
When solving $|ax| + b = c$, $b$ has to be subtracted from both sides first to isolate the absolute value expression.

• Explanation: In the equation $|ax| + b = c$, you first subtract $b$ from both sides to isolate the absolute value, resulting in $|ax| = c - b$. From there, you solve for $x$. On the other hand, in $|ax + b| = c$, the absolute value expression is already isolated, so you immediately consider the two cases where $ax + b = c$ and $ax + b = -c$ to solve for $x$.

Part 2

b. How is solving $|ax| + b \leq c$ different from solving $|ax + b| \geq c$?

Explanation:

• For $|ax| + b \leq c$, you first need to subtract $b$ from both sides to isolate the absolute value term, giving $|ax| \leq c - b$. After this, you consider the case where $- (c - b) \leq ax \leq c - b$.

• For $|ax + b| \geq c$, the absolute value expression is already isolated, and you handle it by considering the two cases:

  1. $ax + b \geq c$
  2. $ax + b \leq -c$

These lead to two separate inequalities for $x$.

Follow-Up Questions:

1. What are the general steps for solving an inequality involving an absolute value?
2. How would the solution change if $c$ were negative in $|ax + b| = c$?
3. What does the solution look like when $c = 0$ in $|ax + b| = c$?
4. How do you interpret the results when the absolute value inequality has no solution?
5. How would you solve the equation if the absolute value term is greater than a constant?

Tip: Always ensure the absolute value expression is isolated before breaking it into the two corresponding equations or inequalities.