1.Solve the equation for all values of x.
open vertical bar, 4, x, minus, 1, close vertical bar, plus, 6, equals, 3, x
∣4x−1∣+6=
3x                       

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.

Solve the equation for all values of x: |4x - 1| + 6 = 3x

This page provides a step-by-step solution to the absolute value equation |4x - 1| + 6 = 3x. It explains how to isolate the absolute value, solve for both positive and negative cases, and check the solutions. Suitable for students in Grades 9-11.

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