What are the properties of linear equations, intercepts, inequalities, and solving equations based on a graph?

The graph of a linear equation is a straight line.

Answer: straight line

The x-intercept is the point where the line crosses the x-axis (where $y=0$). From the graph, the line crosses the x-axis at $x = -2$.
The y-intercept is the point where the line crosses the y-axis (where $x=0$). From the graph, the line crosses the y-axis at $y = -1$.

Answer: x-intercept: $(-2, 0)$, y-intercept: $(0, -1)$

The shaded region is bounded by the x-axis ($y=0$), the y-axis ($x=0$), and a diagonal line.
The diagonal line passes through $(0, 12)$ and $(6, 0)$.
First, find the slope of the line:

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 12}{6 - 0} = \frac{-12}{6} = -2$$

Next, use the point-slope form $y - y_1 = m(x - x_1)$ with point $(6, 0)$:

$$y - 0 = -2(x - 6)$$ $$y = -2x + 12$$

Rearrange to standard form:

$$2x + y = 12$$

The shaded region is below this line. Test a point in the shaded region, e.g., $(1, 1)$:

$$2(1) + 1 = 3$$

Since $3 < 12$, the inequality is $2x + y \le 12$.
The region is also to the right of the y-axis, so $x \ge 0$.
The region is also above the x-axis, so $y \ge 0$.

Answer: $\boxed{2x + y \le 12, x \ge 0, y \ge 0}$

Let the first integer be $n$.
The next consecutive integer is $n+1$.
The sum of these two integers is 47.

$$n + (n+1) = 47$$ $$2n + 1 = 47$$

Subtract 1 from both sides:

$$2n = 47 - 1$$ $$2n = 46$$

Divide by 2:

$$n = \frac{46}{2}$$ $$n = 23$$

The first integer is 23.
The second integer is $n+1 = 23+1 = 24$.

Answer: The integers are $\mathbf{23}$ and $\mathbf{24}$.