1. **State the problem:** We need to analyze and graph the function $$y = |x| - |x - 1|$$. 2. **Recall the definition of absolute value:** For any real number $a$, $$|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$$ 3. **Break the function into intervals based on the critical points where the expressions inside the absolute values change sign:** These points are $x=0$ and $x=1$. 4. **Evaluate $y$ on each interval:** - For $x < 0$: $$|x| = -x, \quad |x-1| = 1 - x$$ So, $$y = -x - (1 - x) = -x - 1 + x = -1$$ - For $0 \leq x < 1$: $$|x| = x, \quad |x-1| = 1 - x$$ So, $$y = x - (1 - x) = x - 1 + x = 2x - 1$$ - For $x \geq 1$: $$|x| = x, \quad |x-1| = x - 1$$ So, $$y = x - (x - 1) = x - x + 1 = 1$$ 5. **Summarize the piecewise function:** $$ y = \begin{cases} -1 & x < 0 \\ 2x - 1 & 0 \leq x < 1 \\ 1 & x \geq 1 \end{cases} $$ 6. **Interpretation:** - For $x < 0$, the function is constant at $-1$. - Between $0$ and $1$, the function is a line with slope $2$ and intercept $-1$. - For $x \geq 1$, the function is constant at $1$. 7. **Final answer:** The function is piecewise linear with the above definition.