Question

find\:the\:inequality\left(0,\:6\right),\:\left(2,\:4\right),\:\left(4,\:2\right),\:\left(5,\:1\right)

Answer 1

To find the inequality that represents the given points, we need to find the slope of the line passing through these points.

The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Let's calculate the slopes for the given points:

For the points $(0, 6)$ and $(2, 4)$, the slope is:
\[m_1 = \frac{4 - 6}{2 - 0} = -1\]

For the points $(2, 4)$ and $(4, 2)$, the slope is:
\[m_2 = \frac{2 - 4}{4 - 2} = -1\]

For the points $(4, 2)$ and $(5, 1)$, the slope is:
\[m_3 = \frac{1 - 2}{5 - 4} = -1\]

Since all three slopes are the same, we can conclude that the inequality that represents these points is of the form:
\[y < mx + n\]

where $m$ is the slope and $n$ is the y-intercept. Since the slope is $-1$, the inequality is of the form:
\[y < -x + n\]

To find the value of $n$, we can substitute any of the given points into the inequality. Let's substitute $(2, 4)$:
\[4 < -(2) + n\]
\[4 < -2 + n\]
\[6 < n\]

Hence, the inequality that represents the given points is:
\[y < -x + 6\]