Solving Equations with Absolute Values: A Comprehensive Guide

When dealing with equations involving absolute values, it's essential to understand the implications of the absolute value function. This article will walk you through the process of solving equations such as 2x - 1 x - 5, systematically breaking it down into manageable cases. By the end, you will understand how to handle different scenarios and find the correct solutions.

Introduction to Absolute Value Equations

Equations involving absolute values can often be solved by considering different cases based on the value of the expression inside the absolute value. The absolute value function, denoted as |x|, is defined as:

|x| x if x ≥ 0 |x| -x if x

By applying these definitions, we can rewrite and solve the equation in a way that covers all possible values of the variable under the absolute value.

Solving the Equation 2x - 1 x - 5

To solve the equation 2x - 1 x - 5, we need to consider the definition of the absolute value, which leads to two cases.

Case 1: x - 5 ≥ 0 (x ≥ 5)

In this case, the absolute value can be removed without changing the sign:

2x - 1 x - 5

Now, solve for x:

2x - 1 x - 5

2x - x -5 1

x -4

Since -4 is not ≤ 5, this solution does not satisfy the condition x ≥ 5 for this case.

Case 2: x - 5

In this case, the absolute value changes the sign:

2x - 1 -x - 5

Now, solve for x:

2x - 1 -x - 5

2x x -5 1

3x -4

x 2

Since 2 is

Conclusion

The only value of x that satisfies the equation 2x - 1 x - 5 is:

boxed{2}

Graphical Interpretation

A graphical representation of the solution can help visualize the relationship between the variables. The plot of the equation 2x - 1 x - 5 shows that the only intersection point, where the two lines meet, is at x 2.

Advanced Techniques

For more complex equations, such as 2x - 1 ±(x - 5), we need to consider all possible cases:

Case 1: x - 5 ≥ 0 (x ≥ 5)

2x - 1 x - 5

Solving for x:

2x - 1 x - 5

2x - x -5 1

x -4

Since x -4 is not ≥ 5, this solution is not valid.

Case 2: x - 5

2x - 1 -(x - 5)

Solving for x:

2x - 1 -x 5

2x x 5 1

3x 6

x 2

Since x 2 is

Final Conclusion

The only solution is:

x 2

This article has covered the essential techniques for solving absolute value equations, providing a clear and structured approach to tackling such problems.