1. **State the problem:** Simplify and analyze the expression $X - |x| - 24$. 2. **Recall the absolute value definition:** The absolute value $|x|$ is defined as: $$ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} $$ 3. **Rewrite the expression based on the sign of $x$:** - If $x \geq 0$, then $|x| = x$, so the expression becomes: $$ X - x - 24 $$ - If $x < 0$, then $|x| = -x$, so the expression becomes: $$ X - (-x) - 24 = X + x - 24 $$ 4. **Interpretation:** The expression depends on the value of $x$: - For $x \geq 0$, it simplifies to $X - x - 24$. - For $x < 0$, it simplifies to $X + x - 24$. 5. **Summary:** The expression $X - |x| - 24$ can be written as a piecewise function: $$ f(x) = \begin{cases} X - x - 24 & \text{if } x \geq 0 \\ X + x - 24 & \text{if } x < 0 \end{cases} $$ This form helps understand how the expression behaves depending on the sign of $x$.