The given inequality is:

\[
3 |x| \leq |x - 1| + 5
\]

To solve this, we need to consider the behavior of the absolute value function. Absolute value functions have different cases based on the sign of the expression inside them, so we will break the problem down into intervals and solve for each case.

### Case 1: \( x \geq 1 \)
- In this case, both \( |x| = x \) and \( |x - 1| = x - 1 \).

Substituting these into the inequality:
\[
3x \leq (x - 1) + 5
\]
Simplifying:
\[
3x \leq x + 4
\]
Subtract \( x \) from both sides:
\[
2x \leq 4
\]
Divide by 2:
\[
x \leq 2
\]

In this case, \( x \geq 1 \) and \( x

Learn how to solve the absolute value inequality 3|x| ≤ |x - 1| + 5 with a detailed step-by-step solution. This algebraic problem involves analyzing absolute value functions and solving linear inequalities, suitable for students in Grades 9-11.

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