Write the four inequalities that define the unshaded region R on the XOY plane below.

Question

ScanSolve AI Tutor · Accepted Answer

let's define the unshaded region R with inequalities.

To define the unshaded region R, we need to identify the four boundary lines and determine the correct inequality for each.

Identify the vertical boundary line.
There is a solid vertical line at $x = 2$ . The unshaded region R is to the right of this line.
The inequality is $x \ge 2$ .
Identify the horizontal boundary lines.
There is a solid horizontal line at $y = 5$ . The unshaded region R is below this line.
The inequality is $y \le 5$ .
There is another solid horizontal line at $y = -1$ . The unshaded region R is above this line.
The inequality is $y \ge -1$ .
Identify the slanted boundary line.
This solid line passes through the points $(0, 9)$ and $(9, 0)$ . We can find its equation.
The slope $m = \frac{0 - 9}{9 - 0} = \frac{-9}{9} = -1$ .
The y-intercept is 9.
So, the equation of the line is $y = -x + 9$ , which can be rewritten as $x + y = 9$ .
The unshaded region R is below this line. To confirm the inequality, pick a point in region R, for example, $(4, 2)$ .
Substitute into $x + y$ : $4 + 2 = 6$ . Since $6 < 9$ , the inequality is $x + y \le 9$ .

The four inequalities that define the unshaded region R are:
$\boxed{x \ge 2}$ $\boxed{y \le 5}$ $\boxed{y \ge -1}$ $\boxed{x + y \le 9}$

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