Solved on Dec 02, 2023

Determine if the solution region for the given system of linear inequalities is bounded or unbounded.
$-x + 2y \geq 4, 2x - y \leq 2, x \geq 0, y \geq 0$

STEP 1

Assumptions
1. The inequalities given are: - $-x+2y \geq 4$ - $2x-y \leq 2$ - $x \geq 0$ - $y \geq 0$
2. We are asked to determine whether the solution region for these inequalities is bounded or unbounded.

STEP 2

Let's first rewrite each inequality in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
For $-x+2y \geq 4$ , we get:
$y \geq \frac{1}{2}x + 2$
For $2x-y \leq 2$ , we get:
$y \geq 2x - 2$

STEP 3

Now, we need to graph these inequalities along with $x \geq 0$ and $y \geq 0$ .
Note: This step is not possible to show in LaTeX, as LaTeX is a typesetting system and does not support graphing. However, you can use any graphing tool to graph these inequalities.

STEP 4

After graphing, we need to identify the solution region. This is the region where all the inequalities overlap.

STEP 5

If the solution region is enclosed by the lines and does not extend to infinity, it is bounded. If the solution region extends to infinity, it is unbounded.

STEP 6

From the graph, we can see that the solution region is enclosed and does not extend to infinity. Therefore, the solution region is bounded.
The solution region is bounded.

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Determine if the solution region for the given system of linear inequalities is bounded or unbounded.
$-x + 2y \geq 4, 2x - y \leq 2, x \geq 0, y \geq 0$

STEP 1

Assumptions
1. The inequalities given are: - $-x+2y \geq 4$ - $2x-y \leq 2$ - $x \geq 0$ - $y \geq 0$
2. We are asked to determine whether the solution region for these inequalities is bounded or unbounded.

STEP 2

Let's first rewrite each inequality in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
For $-x+2y \geq 4$ , we get:
$y \geq \frac{1}{2}x + 2$
For $2x-y \leq 2$ , we get:
$y \geq 2x - 2$

STEP 3

Now, we need to graph these inequalities along with $x \geq 0$ and $y \geq 0$ .
Note: This step is not possible to show in LaTeX, as LaTeX is a typesetting system and does not support graphing. However, you can use any graphing tool to graph these inequalities.

STEP 4

After graphing, we need to identify the solution region. This is the region where all the inequalities overlap.

STEP 5

If the solution region is enclosed by the lines and does not extend to infinity, it is bounded. If the solution region extends to infinity, it is unbounded.

STEP 6

From the graph, we can see that the solution region is enclosed and does not extend to infinity. Therefore, the solution region is bounded.
The solution region is bounded.

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Determine if the solution region for the given system of linear inequalities is bounded or unbounded.−x+2y≥4,2x−y≤2,x≥0,y≥0-x + 2y \geq 4, 2x - y \leq 2, x \geq 0, y \geq 0−x+2y≥4,2x−y≤2,x≥0,y≥0

STEP 1

Assumptions1. The inequalities given are: - −x+2y≥4-x+2y \geq 4−x+2y≥4 - 2x−y≤22x-y \leq 22x−y≤2 - x≥0x \geq 0x≥0 - y≥0y \geq 0y≥02. We are asked to determine whether the solution region for these inequalities is bounded or unbounded.

STEP 2

Let's first rewrite each inequality in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.For −x+2y≥4-x+2y \geq 4−x+2y≥4, we get:y≥12x+2y \geq \frac{1}{2}x + 2y≥21​x+2For 2x−y≤22x-y \leq 22x−y≤2, we get:y≥2x−2y \geq 2x - 2y≥2x−2

STEP 3

Now, we need to graph these inequalities along with x≥0x \geq 0x≥0 and y≥0y \geq 0y≥0.Note: This step is not possible to show in LaTeX, as LaTeX is a typesetting system and does not support graphing. However, you can use any graphing tool to graph these inequalities.

STEP 4

After graphing, we need to identify the solution region. This is the region where all the inequalities overlap.

STEP 5

If the solution region is enclosed by the lines and does not extend to infinity, it is bounded. If the solution region extends to infinity, it is unbounded.

STEP 6

From the graph, we can see that the solution region is enclosed and does not extend to infinity. Therefore, the solution region is bounded.The solution region is bounded.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Determine if the solution region for the given system of linear inequalities is bounded or unbounded.
$-x + 2y \geq 4, 2x - y \leq 2, x \geq 0, y \geq 0$

Assumptions
1. The inequalities given are: - $-x+2y \geq 4$ - $2x-y \leq 2$ - $x \geq 0$ - $y \geq 0$
2. We are asked to determine whether the solution region for these inequalities is bounded or unbounded.

Let's first rewrite each inequality in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
For $-x+2y \geq 4$ , we get:
$y \geq \frac{1}{2}x + 2$
For $2x-y \leq 2$ , we get:
$y \geq 2x - 2$

Now, we need to graph these inequalities along with $x \geq 0$ and $y \geq 0$ .
Note: This step is not possible to show in LaTeX, as LaTeX is a typesetting system and does not support graphing. However, you can use any graphing tool to graph these inequalities.

From the graph, we can see that the solution region is enclosed and does not extend to infinity. Therefore, the solution region is bounded.
The solution region is bounded.