Math / Algebra

Question $\begin{array}{l}x+y \geq 2 \\ 4 x+y \geq-1\end{array}$

Studdy Solution

STEP 1

1. We are given two linear inequalities.
2. We need to find the region in the coordinate plane that satisfies both inequalities simultaneously.
3. The solution will be a region, not a single point.

STEP 2

1. Graph the line $x + y = 2$ and determine the region for $x + y \geq 2$ .
2. Graph the line $4x + y = -1$ and determine the region for $4x + y \geq -1$ .
3. Find the intersection of the two regions.

STEP 3

Graph the line $x + y = 2$ :
- Rearrange the equation to solve for $y$ : $y = -x + 2$ - This is a line with a slope of $-1$ and a y-intercept of $2$ . - Plot the line on the coordinate plane.

STEP 4

Determine the region for $x + y \geq 2$ :
- The inequality $x + y \geq 2$ means the region above or on the line $y = -x + 2$ . - Shade the region above the line, including the line itself.

STEP 5

Graph the line $4x + y = -1$ :
- Rearrange the equation to solve for $y$ : $y = -4x - 1$ - This is a line with a slope of $-4$ and a y-intercept of $-1$ . - Plot the line on the coordinate plane.

STEP 6

Determine the region for $4x + y \geq -1$ :
- The inequality $4x + y \geq -1$ means the region above or on the line $y = -4x - 1$ . - Shade the region above the line, including the line itself.

STEP 7

Find the intersection of the two regions:
- The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. - Identify and shade this overlapping region on the coordinate plane.
The solution is the region of the coordinate plane where both shaded areas intersect, representing all points $(x, y)$ that satisfy both inequalities simultaneously.

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Question $\begin{array}{l}x+y \geq 2 \\ 4 x+y \geq-1\end{array}$

Studdy Solution

STEP 1

1. We are given two linear inequalities.
2. We need to find the region in the coordinate plane that satisfies both inequalities simultaneously.
3. The solution will be a region, not a single point.

STEP 2

1. Graph the line $x + y = 2$ and determine the region for $x + y \geq 2$ .
2. Graph the line $4x + y = -1$ and determine the region for $4x + y \geq -1$ .
3. Find the intersection of the two regions.

STEP 3

Graph the line $x + y = 2$ :
- Rearrange the equation to solve for $y$ : $y = -x + 2$ - This is a line with a slope of $-1$ and a y-intercept of $2$ . - Plot the line on the coordinate plane.

STEP 4

Determine the region for $x + y \geq 2$ :
- The inequality $x + y \geq 2$ means the region above or on the line $y = -x + 2$ . - Shade the region above the line, including the line itself.

STEP 5

Graph the line $4x + y = -1$ :
- Rearrange the equation to solve for $y$ : $y = -4x - 1$ - This is a line with a slope of $-4$ and a y-intercept of $-1$ . - Plot the line on the coordinate plane.

STEP 6

Determine the region for $4x + y \geq -1$ :
- The inequality $4x + y \geq -1$ means the region above or on the line $y = -4x - 1$ . - Shade the region above the line, including the line itself.

STEP 7

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Questionx+y≥24x+y≥−1\begin{array}{l}x+y \geq 2 \\ 4 x+y \geq-1\end{array}x+y≥24x+y≥−1​

Studdy Solution

STEP 1

1. We are given two linear inequalities.2. We need to find the region in the coordinate plane that satisfies both inequalities simultaneously.3. The solution will be a region, not a single point.

STEP 2

1. Graph the line x+y=2x + y = 2x+y=2 and determine the region for x+y≥2x + y \geq 2x+y≥2.2. Graph the line 4x+y=−14x + y = -14x+y=−1 and determine the region for 4x+y≥−14x + y \geq -14x+y≥−1.3. Find the intersection of the two regions.

STEP 3

Graph the line x+y=2x + y = 2x+y=2:- Rearrange the equation to solve for yyy: y=−x+2 y = -x + 2 y=−x+2 - This is a line with a slope of −1-1−1 and a y-intercept of 222. - Plot the line on the coordinate plane.

STEP 4

Determine the region for x+y≥2x + y \geq 2x+y≥2:- The inequality x+y≥2x + y \geq 2x+y≥2 means the region above or on the line y=−x+2y = -x + 2y=−x+2. - Shade the region above the line, including the line itself.

STEP 5

Graph the line 4x+y=−14x + y = -14x+y=−1:- Rearrange the equation to solve for yyy: y=−4x−1 y = -4x - 1 y=−4x−1 - This is a line with a slope of −4-4−4 and a y-intercept of −1-1−1. - Plot the line on the coordinate plane.

STEP 6

Determine the region for 4x+y≥−14x + y \geq -14x+y≥−1:- The inequality 4x+y≥−14x + y \geq -14x+y≥−1 means the region above or on the line y=−4x−1y = -4x - 1y=−4x−1. - Shade the region above the line, including the line itself.

STEP 7

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Question $\begin{array}{l}x+y \geq 2 \\ 4 x+y \geq-1\end{array}$

1. We are given two linear inequalities.
2. We need to find the region in the coordinate plane that satisfies both inequalities simultaneously.
3. The solution will be a region, not a single point.

1. Graph the line $x + y = 2$ and determine the region for $x + y \geq 2$ .
2. Graph the line $4x + y = -1$ and determine the region for $4x + y \geq -1$ .
3. Find the intersection of the two regions.

Graph the line $x + y = 2$ :
- Rearrange the equation to solve for $y$ : $y = -x + 2$ - This is a line with a slope of $-1$ and a y-intercept of $2$ . - Plot the line on the coordinate plane.

Determine the region for $x + y \geq 2$ :
- The inequality $x + y \geq 2$ means the region above or on the line $y = -x + 2$ . - Shade the region above the line, including the line itself.

Graph the line $4x + y = -1$ :
- Rearrange the equation to solve for $y$ : $y = -4x - 1$ - This is a line with a slope of $-4$ and a y-intercept of $-1$ . - Plot the line on the coordinate plane.

Determine the region for $4x + y \geq -1$ :
- The inequality $4x + y \geq -1$ means the region above or on the line $y = -4x - 1$ . - Shade the region above the line, including the line itself.