Question: User: Graph: a downward-sloping straight line with arrowheads, crossing the y-axis at about $7$ and the x-axis at about $4.5$; the shaded region is on the lower-left side of the line, and there are two filled points on the line near $(0, 7)$ and $(0, 4)$, plus one filled point near $(3, 0)$. position_hint=top-leftShown here is the graph of the region described by which set of inequalities?
1. **State the problem:** We are given a graph of a downward-sloping line crossing the y-axis at about $7$ and the x-axis at about $4.5$, with the shaded region on the lower-left side of the line. We need to find the set of inequalities describing this region.
2. **Find the equation of the line:** The line crosses the y-axis at $y=7$, so the y-intercept $b=7$.
The x-intercept is about $4.5$, so the line passes through $(4.5, 0)$.
The slope $m$ is given by:
$$m = \frac{0 - 7}{4.5 - 0} = \frac{-7}{4.5} = -\frac{14}{9}$$
3. **Write the line equation:** Using slope-intercept form:
$$y = mx + b = -\frac{14}{9}x + 7$$
4. **Determine the inequality:** The shaded region is on the lower-left side of the line, which means the region below the line.
So the inequality is:
$$y \leq -\frac{14}{9}x + 7$$
5. **Check points:** The point $(0,4)$ is on the line (approximately), and $(3,0)$ is in the shaded region.
Check $(3,0)$:
$$0 \leq -\frac{14}{9} \times 3 + 7 = -\frac{42}{9} + 7 = -4.666... + 7 = 2.333...$$
Since $0 \leq 2.333...$ is true, the inequality holds.
**Final answer:** The region is described by the inequality
$$y \leq -\frac{14}{9}x + 7$$