1. **State the problem:** We are given a relation $R$ defined by the inequalities $y \leq x - 2$ and $y \geq 2x - 4$. We need to sketch the graph of this relation. 2. **Understand the inequalities:** - The first inequality $y \leq x - 2$ represents all points on or below the line $y = x - 2$. - The second inequality $y \geq 2x - 4$ represents all points on or above the line $y = 2x - 4$. 3. **Graph the boundary lines:** - For $y = x - 2$, the line has slope 1 and y-intercept $-2$. - For $y = 2x - 4$, the line has slope 2 and y-intercept $-4$. 4. **Find the intersection point of the two lines:** Set $x - 2 = 2x - 4$ to find where the lines meet. $$x - 2 = 2x - 4$$ $$-2 + 4 = 2x - x$$ $$2 = x$$ Substitute $x=2$ into $y = x - 2$: $$y = 2 - 2 = 0$$ So, the lines intersect at the point $(2, 0)$. 5. **Determine the region satisfying both inequalities:** - The region is the set of points $y$ such that $2x - 4 \leq y \leq x - 2$. - This region lies between the two lines, including the lines themselves. 6. **Summary:** - The graph is the shaded region between the lines $y = 2x - 4$ and $y = x - 2$, including the boundary lines. **Final answer:** The graph of the relation $R$ is the region bounded by and including the lines $y = x - 2$ and $y = 2x - 4$, with the intersection point at $(2,0)$.