Question

question 14

solve the equation \\(|5x - 4| - 2 = 2\\)

the solutions are:

question 15

let \\(f(x) = 3x + 11\\)

\\(f^{-1}(x) =\\)

Explanation:

🆕 New Concept Discovered: Solving Absolute Value Equations
Find where the inside splits into positive and negative cases.

Step 1: Isolate the absolute value expression

To solve the equation, we first want to get the absolute value term by itself on one side.

Given:

$$ |5x - 4| - 2 = 2 $$

Add \(2\) to both sides:

$$ |5x - 4| = 4 $$

Step 2: Split into two cases

An absolute value equation \(|u| = a\) (where \(a \ge 0\)) means the expression inside, \(u\), can be either \(a\) or \(-a\). This is because both numbers are the same distance from zero.

This gives us two separate equations to solve:

1. Case 1: \(5x - 4 = 4\)
2. Case 2: \(5x - 4 = -4\)

Step 3: Solve Case 1

$$ 5x - 4 = 4 $$

Add \(4\) to both sides:

$$ 5x = 8 $$

Divide by \(5\):

$$ x = \frac{8}{5} $$

Step 4: Solve Case 2

$$ 5x - 4 = -4 $$

Add \(4\) to both sides:

$$ 5x = 0 $$

Divide by \(5\):

$$ x = 0 $$

Answer:

\(x = 0, \frac{8}{5}\)