It looks like you're trying to solve the inequality involving the absolute value:

$|4x - 1| - x + 2 \geq 0$

To solve this inequality, we need to break it into cases based on the behavior of the absolute value.

Step 1: Consider two cases for $|4x - 1|$

1. Case 1: $4x - 1 \geq 0$, or $x \geq \frac{1}{4}$ In this case, $|4x - 1| = 4x - 1$. The inequality becomes:

   $(4x - 1) - x + 2 \geq 0$

   Simplifying:

   $4x - 1 - x + 2 \geq 0$ $3x + 1 \geq 0$ $3x \geq -1$ $x \geq -\frac{1}{3}$

   Since this case assumes $x \geq \frac{1}{4}$, the solution for this case is:

   $x \geq \frac{1}{4}$

2. Case 2: $4x - 1 < 0$, or $x < \frac{1}{4}$ In this case, $|4x - 1| = -(4x - 1) = -4x + 1$. The inequality becomes:

   $(-4x + 1) - x + 2 \geq 0$

   Simplifying:

   $-4x + 1 - x + 2 \geq 0$ $-5x + 3 \geq 0$ $-5x \geq -3$ $x \leq \frac{3}{5}$

   Since this case assumes $x < \frac{1}{4}$, the solution for this case is:

   $x < \frac{1}{4}$

Step 2: Combine the results

• From Case 1, we found $x \geq \frac{1}{4}$.
• From Case 2, we found $x < \frac{1}{4}$.

Thus, the combined solution is the union of both cases:

$x \in (-\infty, \frac{3}{5}]$

Final Answer:

The solution to the inequality $|4x - 1| - x + 2 \geq 0$ is:

$x \in \left(-\infty, \frac{3}{5}\right]$

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Would you like further details or have any specific questions? Here's a list of related questions to deepen the understanding:

1. How does the absolute value affect inequalities and their solutions?
2. What are the general steps to solve absolute value inequalities?
3. Can inequalities with absolute values have multiple ranges of solutions?
4. How can we graph the solution of an absolute value inequality?
5. What is the role of boundary points in absolute value inequalities?

Tip: Always break absolute value inequalities into cases based on the sign of the expression inside the absolute value!