Solving Inequalities with Absolute Values: A Comprehensive Guide

When tackling inequalities that involve absolute values, it's crucial to understand the behavior of the expressions within these absolute values. This guide will walk you through the process of solving an inequality such as x - 6 |x - 1| ≤ 11. We will explore the method of breaking down the problem into manageable intervals and using critical points to solve it accurately.

Understanding the Critical Points

First, identify the critical points where the expressions inside the absolute values change their signs. In our inequality, the critical points are x 1 and x 6. These points divide the real number line into three distinct intervals: (-∞, 1), [1, 6], and (6, ∞).

Dividing Intervals Based on Critical Points

Interval 1: (-∞, 1)

In this interval, both expressions x - 6 and x - 1 are negative. To handle the absolute values, rewrite them as:

|x - 6| 6 - x and |x - 1| 1 - x

The inequality x - 6 |x - 1| ≤ 11 becomes:

(6 - x) (1 - x) ≤ 11

Simplifying this, we get:

7 - 2x ≤ 11

Subtract 7 from both sides:

-2x ≤ 4

Dividing by -2 and reversing the inequality:

x ≥ -2

So, for this interval, the solution is:

-2 ≤ x ≤ 1

Interval 2: [1, 6]

In this interval, the expression x - 6 is negative, and x - 1 is positive. Therefore, we rewrite the absolute values as:

|x - 6| 6 - x and |x - 1| x - 1

The inequality becomes:

(6 - x) - (x - 1) ≤ 11

Simplifying this, we get:

6 - x - x   1 ≤ 115 ≤ 11

This statement is always true for all x in the interval [1, 6]. Therefore, the solution for this interval is:

1 ≤ x ≤ 6

Interval 3: (6, ∞)

In this interval, both expressions x - 6 and x - 1 are positive. The absolute values are:

|x - 6| x - 6 and |x - 1| x - 1

The inequality becomes:

(x - 6) (x - 1) ≤ 11

Simplifying this, we get:

2x - 7 ≤ 112x ≤ 18x ≤ 9

So, for this interval, the solution is:

6 ≤ x ≤ 9

Combining the Intervals

To find the overall solution to the inequality x - 6 |x - 1| ≤ 11, we combine the solutions from all intervals:

-2 ≤ x ≤ 11 ≤ x ≤ 66 ≤ x ≤ 9

Merging these intervals, we obtain the final solution:

-2 ≤ x ≤ 9

Conclusion

The solution to the inequality x - 6 |x - 1| ≤ 11 is:

[-2, 9]

This method of solving absolute value inequalities by dividing the number line into intervals using critical points and then solving each segment is a powerful tool in algebra and is widely applicable to various problem types. Understanding these steps is key to solving more complex inequalities involving absolute values.