Let a, b, c and x be real numbers. How is solving  different from solving 

The correct answer is: -(b+c)/a≤x and x≥-(b-c)/a Finally, we can observe that the solutions we get from the two given inequalities are different.

Hint:
|x| is known as the absolute value of x. It is the non-negative value of x irrespective of its sign. The value of absolute value of x is given by

The symbol |.| is pronounced as ‘modulus’. We read |x| as ‘modulus of x’ or ‘mod x’.
We discuss both these cases for the two given inequalities.
Step by step solution:
We assume that a, b, c are positive.
The first inequality is
|ax |+ b ≥ c
Applying the definition of the modulus for ax in the above inequality, we get
For ax < 0, we have
-ax + b ≥ c
⇒-ax ≥ c - b
⇒ax ≤ b - c
Then solving for x, we get

For ax ≥ 0, we have
ax + b ≥ c
Solving for x, we get

Or

Combining the two solutions, we have

The second inequality is
|ax + b| ≥ c
We use the definition of modulus,
For ax + b < 0, we have
-(ax + b) ≥ c
⇒-ax - b ≥ c
⇒-ax ≥ b + c
Solving for x, we get

For ax + b ≥ 0, we have
ax + b ≥ c
⇒ax ≥ c - b
Solving for x, we get

Combining the two solutions we get

Finally, we can observe that the solutions we get from the two given inequalities are different.
Note:
In the first inequalities, mod changes the value of only one term, i.e., ax; whereas in the second inequality, mod changes the value two terms, ax and b. This results in the change in values of x for both inequalities.