Certainly! Here are some challenging questions on absolute value, designed to test a deep understanding of absolute values, properties, and applications.

1. Nested Absolute Value Equations

Solve for all real values of $x$: $\left| \left| x - 3 \right| - 2 \right| = 5$

2. Absolute Value Inequality with Quadratics

Determine the set of all $x$ that satisfy: $\left| x^2 - 5x + 6 \right| \leq 4$

3. System of Absolute Value Equations

Solve the system of equations:

\left| x + y \right| = 7 \\ \left| x - y \right| = 3 \end{cases}$$ ### 4. Absolute Value Function with Parameter For what values of $$k$$ does the equation $$\left| x^2 - kx + k - 4 \right| = k$$ have exactly three distinct real solutions? ### 5. Integral of an Absolute Value Function Evaluate the integral: $$\int_{-2}^{4} \left| x^2 - 3x \right| \, dx$$ These problems cover various aspects of absolute value, including nested expressions, inequalities, systems, parameter dependencies, and integration involving absolute value functions. Would you like any of these problems explained in detail, or help in solving one of them? --- **Related Questions for Deeper Understanding:** 1. How do you solve absolute value equations with nested absolute values? 2. What is the method to approach absolute value inequalities involving quadratic functions? 3. How do systems of equations with absolute values differ from linear systems? 4. What are some properties of absolute value that affect the number of solutions in an equation? 5. How do absolute values affect the calculation of integrals? **Tip:** When solving absolute value equations or inequalities, consider breaking them down by analyzing different cases based on the intervals where the expressions inside the absolute values change sign.