Solve the following inequalities and represent the solution on a number line. a) y >= x - 1 b) 2x - y <= 4

The boundary line is given by the equation $y = x - 1$.

• If $x = 0$, then $y = 0 - 1 = -1$. So, the point is $(0, -1)$.
• If $y = 0$, then $0 = x - 1 \implies x = 1$. So, the point is $(1, 0)$.

Plot the points $(0, -1)$ and $(1, 0)$ on the graph paper. Since the inequality is $y \ge x - 1$ (which includes "equal to"), draw a solid line connecting these two points.

Choose a test point not on the line, for example, the origin $(0, 0)$.
Substitute $(0, 0)$ into the inequality $y \ge x - 1$:

This statement is true. Therefore, shade the region that contains the test point $(0, 0)$. This will be the region above the line $y = x - 1$.

The solution is the shaded region including the solid line $y = x - 1$.

Multiply both sides by $-1$ and reverse the inequality sign:

The boundary line is given by the equation $y = 2x - 4$.

• If $x = 0$, then $y = 2(0) - 4 = -4$. So, the point is $(0, -4)$.
• If $y = 0$, then $0 = 2x - 4 \implies 2x = 4 \implies x = 2$. So, the point is $(2, 0)$.

Plot the points $(0, -4)$ and $(2, 0)$ on the graph paper. Since the inequality is $y \ge 2x - 4$ (which includes "equal to"), draw a solid line connecting these two points.

Choose a test point not on the line, for example, the origin $(0, 0)$.
Substitute $(0, 0)$ into the inequality $y \ge 2x - 4$:

This statement is true. Therefore, shade the region that contains the test point $(0, 0)$. This will be the region above the line $y = 2x - 4$.

The solution is the shaded region including the solid line $y = 2x - 4$.