Math / Geometry

Question16 Beskriv det färgade området med en olikhet.

Studdy Solution

STEP 1

1. The coordinate plane is a standard Cartesian plane with x-axis and y-axis.
2. The line with a negative slope passes through the points (0,15) and (15,0).
3. The shaded area is below the line and above the x-axis and y-axis.

STEP 2

1. Determine the equation of the line.
2. Write the inequality that represents the shaded area.

STEP 3

To find the equation of the line, we need the slope and the y-intercept. The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Using the points (0,15) and (15,0):
$m = \frac{0 - 15}{15 - 0} = \frac{-15}{15} = -1$
The y-intercept $b$ is the y-coordinate of the point where the line crosses the y-axis, which is 15.
Thus, the equation of the line is:
$y = -x + 15$

STEP 4

The shaded area is below the line and above the x-axis and y-axis. This means we need an inequality that represents all the points $(x, y)$ that are below the line $y = -x + 15$ and above the axes.
The inequality for the shaded region is:
$y \leq -x + 15$
Additionally, since the shaded region is bounded by the x-axis and y-axis, we also have:
$x \geq 0$ $y \geq 0$
Thus, the complete set of inequalities describing the shaded area is:
$y \leq -x + 15$ $x \geq 0$ $y \geq 0$
The inequality that describes the shaded area is:
$y \leq -x + 15, \quad x \geq 0, \quad y \geq 0$

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