Question

find the real solutions of the equation. |3 - 4t| + 5 = 24 select the correct choice below and fill in any answer boxes within your choice. a. the solution set is { }. (simplify your answer. use a comma to separate answers as needed.) b. there is no real solution.

Explanation:

Step1: Isolate the absolute value

Subtract 5 from both sides of the equation \(|3 - 4t| + 5 = 24\).
\(|3 - 4t| = 24 - 5\)
\(|3 - 4t| = 19\)

Step2: Solve the absolute value equation

The absolute value equation \(|A| = B\) (where \(B\geq0\)) is equivalent to \(A = B\) or \(A = -B\). Here, \(A = 3 - 4t\) and \(B = 19\) (which is greater than 0). So we have two cases:

Case 1: \(3 - 4t = 19\)

Subtract 3 from both sides:
\(-4t = 19 - 3\)
\(-4t = 16\)
Divide both sides by -4:
\(t = \frac{16}{-4}\)
\(t = -4\)

Case 2: \(3 - 4t = -19\)

Subtract 3 from both sides:
\(-4t = -19 - 3\)
\(-4t = -22\)
Divide both sides by -4:
\(t = \frac{-22}{-4}\)
\(t = \frac{11}{2} = 5.5\)

Answer:

The solution set is \(\{-4, \frac{11}{2}\}\) (or \(-4, 5.5\) if decimal form is preferred). So the correct choice is A with the solution set \(\{-4, \frac{11}{2}\}\).