We are given the equation:
$|4x - 1| + 6 = 3x$

Step 1: Isolate the absolute value expression

Subtract 6 from both sides: $|4x - 1| = 3x - 6$

Step 2: Solve for two cases

Since $|4x - 1|$ represents two possible values (the expression inside the absolute value can be positive or negative), we solve for two cases:

Case 1: Positive case

$4x - 1 = 3x - 6$ Subtract $3x$ from both sides: $x - 1 = -6$ Add 1 to both sides: $x = -5$

Case 2: Negative case

$4x - 1 = -(3x - 6)$ Distribute the negative sign: $4x - 1 = -3x + 6$ Add $3x$ to both sides: $7x - 1 = 6$ Add 1 to both sides: $7x = 7$ Divide by 7: $x = 1$

Step 3: Check both solutions

We need to check if both values satisfy the original equation.

For $x = -5$: $|4(-5) - 1| + 6 = 3(-5)$ $|-20 - 1| + 6 = -15$ $| -21 | + 6 = -15$ $21 + 6 \neq -15$ So, $x = -5$ is not a solution.

For $x = 1$: $|4(1) - 1| + 6 = 3(1)$ $|4 - 1| + 6 = 3$ $|3| + 6 = 3$ $3 + 6 \neq 3$ This doesn't satisfy either, so there is no solution to this equation.

Conclusion:

There is no solution to this equation.

Would you like further explanation or more details on absolute value equations?

Here are 5 related questions:

1. What are the properties of absolute value equations?
2. How do you handle equations with two absolute value terms?
3. Can you graph absolute value functions and understand their behavior?
4. How would this change if $3x - 6$ were a quadratic expression?
5. What if the equation involved inequalities instead?

Tip: Always check both solutions when solving absolute value equations to ensure they satisfy the original equation.