1. The problem is to find the inequalities that define the region $R$ bounded by the lines $x=6$, $y=x$, and $y=1$. 2. First, note the lines: - Vertical line: $x=6$ - Diagonal line: $y=x$ - Horizontal line: $y=1$ 3. The region $R$ is bounded between these lines. To describe it with inequalities, consider the following: 4. Since $x=6$ is a vertical boundary, the region lies to the left of this line, so $x \leq 6$. 5. The diagonal line $y=x$ forms a boundary where $y$ is equal to $x$. The region lies above this line, so $y \geq x$. 6. The horizontal line $y=1$ forms the lower boundary, so the region lies above or on this line, $y \geq 1$. 7. Combining these, the inequalities defining region $R$ are: $$ \begin{cases} x \leq 6 \\ y \geq x \\ y \geq 1 \end{cases} $$ 8. This means the region $R$ includes all points $(x,y)$ such that $x$ is at most 6, $y$ is at least $x$, and $y$ is at least 1. Final answer: $$ \boxed{\{(x,y) \mid x \leq 6, y \geq x, y \geq 1\}} $$