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Sagot :
Certainly! Let's solve the given inequality:
[tex]\[ x + 2y \geq 4 \][/tex]
### Step 1: Rewrite the Inequality
To make it easier to understand and graph, let's isolate [tex]\( x \)[/tex]. Begin by manipulating the inequality as follows:
[tex]\[ x \geq 4 - 2y \][/tex]
This tells us that [tex]\( x \)[/tex] must be greater than or equal to [tex]\( 4 - 2y \)[/tex].
### Step 2: Consider the Regions of the Graph
This inequality defines a region on the [tex]\( xy \)[/tex]-plane. Specifically, it indicates that for any given [tex]\( y \)[/tex]-value, [tex]\( x \)[/tex] must be at least [tex]\( 4 - 2y \)[/tex].
### Step 3: Boundary Line
The boundary line of the inequality is given by:
[tex]\[ x = 4 - 2y \][/tex]
This is a straight line, and to graph it, we need a few points:
1. When [tex]\( y = 0 \)[/tex]:
[tex]\[ x = 4 \][/tex]
Thus, the point [tex]\((4, 0)\)[/tex].
2. When [tex]\( y = 2 \)[/tex]:
[tex]\[ x = 4 - 2(2) = 4 - 4 = 0 \][/tex]
Thus, the point [tex]\((0, 2)\)[/tex].
These two points, [tex]\((4, 0)\)[/tex] and [tex]\((0, 2)\)[/tex], help us draw the line [tex]\( x = 4 - 2y \)[/tex].
### Step 4: Graph the Boundary Line
Draw the line through the points [tex]\((4, 0)\)[/tex] and [tex]\((0, 2)\)[/tex] on the [tex]\( xy \)[/tex]-plane.
### Step 5: Determine the Region
Since the inequality is [tex]\( x \geq 4 - 2y \)[/tex], the region of interest is on or to the right of this line. This region includes all points [tex]\((x, y)\)[/tex] where [tex]\( x \)[/tex] is greater than or equal to [tex]\( 4 - 2y \)[/tex].
### Step 6: Verify with a Test Point
Choose a test point not on the line, such as [tex]\((5, 0)\)[/tex]:
1. Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[ 5 \geq 4 - 2(0) \][/tex]
[tex]\[ 5 \geq 4 \][/tex]
This is true, so the point [tex]\((5, 0)\)[/tex] is within the region, confirming that our region is correct.
### Conclusion
Thus, the solution to the inequality [tex]\( x + 2y \geq 4 \)[/tex] involves all points [tex]\((x, y)\)[/tex] that lie on or to the right of the line [tex]\( x = 4 - 2y \)[/tex]. The inequality can be restated as:
[tex]\[ x + 2y - 4 \geq 0 \][/tex]
This inequality defines the region where the sum of [tex]\( x \)[/tex] and twice [tex]\( y \)[/tex] is at least 4.
[tex]\[ x + 2y \geq 4 \][/tex]
### Step 1: Rewrite the Inequality
To make it easier to understand and graph, let's isolate [tex]\( x \)[/tex]. Begin by manipulating the inequality as follows:
[tex]\[ x \geq 4 - 2y \][/tex]
This tells us that [tex]\( x \)[/tex] must be greater than or equal to [tex]\( 4 - 2y \)[/tex].
### Step 2: Consider the Regions of the Graph
This inequality defines a region on the [tex]\( xy \)[/tex]-plane. Specifically, it indicates that for any given [tex]\( y \)[/tex]-value, [tex]\( x \)[/tex] must be at least [tex]\( 4 - 2y \)[/tex].
### Step 3: Boundary Line
The boundary line of the inequality is given by:
[tex]\[ x = 4 - 2y \][/tex]
This is a straight line, and to graph it, we need a few points:
1. When [tex]\( y = 0 \)[/tex]:
[tex]\[ x = 4 \][/tex]
Thus, the point [tex]\((4, 0)\)[/tex].
2. When [tex]\( y = 2 \)[/tex]:
[tex]\[ x = 4 - 2(2) = 4 - 4 = 0 \][/tex]
Thus, the point [tex]\((0, 2)\)[/tex].
These two points, [tex]\((4, 0)\)[/tex] and [tex]\((0, 2)\)[/tex], help us draw the line [tex]\( x = 4 - 2y \)[/tex].
### Step 4: Graph the Boundary Line
Draw the line through the points [tex]\((4, 0)\)[/tex] and [tex]\((0, 2)\)[/tex] on the [tex]\( xy \)[/tex]-plane.
### Step 5: Determine the Region
Since the inequality is [tex]\( x \geq 4 - 2y \)[/tex], the region of interest is on or to the right of this line. This region includes all points [tex]\((x, y)\)[/tex] where [tex]\( x \)[/tex] is greater than or equal to [tex]\( 4 - 2y \)[/tex].
### Step 6: Verify with a Test Point
Choose a test point not on the line, such as [tex]\((5, 0)\)[/tex]:
1. Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[ 5 \geq 4 - 2(0) \][/tex]
[tex]\[ 5 \geq 4 \][/tex]
This is true, so the point [tex]\((5, 0)\)[/tex] is within the region, confirming that our region is correct.
### Conclusion
Thus, the solution to the inequality [tex]\( x + 2y \geq 4 \)[/tex] involves all points [tex]\((x, y)\)[/tex] that lie on or to the right of the line [tex]\( x = 4 - 2y \)[/tex]. The inequality can be restated as:
[tex]\[ x + 2y - 4 \geq 0 \][/tex]
This inequality defines the region where the sum of [tex]\( x \)[/tex] and twice [tex]\( y \)[/tex] is at least 4.
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