To solve the equation involving absolute values: $|6 - x| = |5 + x|$

We need to consider the different cases where the expressions inside the absolute values can change sign. This means we will analyze the equation for the scenarios when $6 - x$ and $5 + x$ are either positive or negative.

Case 1: $6 - x \geq 0$ and $5 + x \geq 0$

This implies: $x \leq 6 \quad \text{and} \quad x \geq -5$ In this range, both expressions are non-negative, so we have: $6 - x = 5 + x$ Solving this equation: $6 - 5 = x + x$ $1 = 2x \implies x = \frac{1}{2}$

Case 2: $6 - x \geq 0$ and $5 + x < 0$

This implies: $x \leq 6 \quad \text{and} \quad x < -5$ However, there is no overlap in these conditions, so this case has no solutions.

Case 3: $6 - x < 0$ and $5 + x \geq 0$

This implies: $x > 6 \quad \text{and} \quad x \geq -5$ Again, there is no overlap in these conditions, so this case also has no solutions.

Case 4: $6 - x < 0$ and $5 + x < 0$

This implies: $x > 6 \quad \text{and} \quad x < -5$ This case is also impossible because the conditions cannot be true simultaneously.

Conclusion

The only solution is: $x = \frac{1}{2}$

Would you like more details or have any questions?

Here are some related questions you might be interested in:

1. What are the steps to graph absolute value functions?
2. How do you determine the intervals where an absolute value expression changes sign?
3. How can we handle equations with multiple absolute values?
4. What are some applications of absolute value equations in real-world problems?
5. How do you solve absolute value inequalities?

Tip: When solving absolute value equations, always consider all possible cases for the signs of the expressions inside the absolute values.