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Sagot :
To determine which statements about the system of inequalities given by [tex]\( y \leq 3x + 1 \)[/tex] and [tex]\( y \geq -x + 2 \)[/tex] are true, let’s analyze each statement one by one.
### 1. "The slope of one boundary line is 2"
To find the slopes of the lines defined by the inequalities:
- For [tex]\( y \leq 3x + 1 \)[/tex]: The inequality is already in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Here, the slope [tex]\( m \)[/tex] is 3.
- For [tex]\( y \geq -x + 2 \)[/tex]: Similarly, this inequality is in slope-intercept form, and the slope [tex]\( m \)[/tex] is -1.
Hence, the slopes of the lines are 3 and -1. Neither of these slopes is 2. Therefore, the statement "The slope of one boundary line is 2" is false.
### 2. "Both boundary lines are solid"
Inequalities using [tex]\( \leq \)[/tex] (less than or equal to) and [tex]\( \geq \)[/tex] (greater than or equal to) dictate that the boundary lines are solid, indicating that the points on these lines are included in the solution set.
Therefore, the statement "Both boundary lines are solid" is true.
### 3. "A solution to the system is [tex]\((1, 3)\)[/tex]"
To check if the point [tex]\((1, 3)\)[/tex] is a solution to both inequalities:
- For [tex]\( y \leq 3x + 1 \)[/tex]: Plug in [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 \leq 3(1) + 1 \][/tex]
[tex]\[ 3 \leq 4 \][/tex]
This holds true.
- For [tex]\( y \geq -x + 2 \)[/tex]: Again, plug in [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 \geq -1 + 2 \][/tex]
[tex]\[ 3 \geq 1 \][/tex]
This also holds true.
Thus, the point [tex]\((1, 3)\)[/tex] satisfies both inequalities. Therefore, the statement "A solution to the system is [tex]\((1, 3)\)[/tex]" is true.
### 4. "Both inequalities are shaded below the boundary lines"
To determine where the areas are shaded relative to the boundary lines:
- For [tex]\( y \leq 3x + 1 \)[/tex]: The shading is below the line because [tex]\( y \)[/tex] is less than or equal to [tex]\( 3x + 1 \)[/tex].
- For [tex]\( y \geq -x + 2 \)[/tex]: The shading is above the line because [tex]\( y \)[/tex] is greater than or equal to [tex]\( -x + 2 \)[/tex].
As the shadings are in opposite directions for each inequality, the statement "Both inequalities are shaded below the boundary lines" is false.
### 5. "The boundary lines intersect"
To find if and where the boundary lines intersect, we set the equations equal to each other and solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ 3x + 1 = -x + 2 \][/tex]
Solving this:
[tex]\[ 3x + x = 2 - 1 \][/tex]
[tex]\[ 4x = 1 \][/tex]
[tex]\[ x = \frac{1}{4} \][/tex]
Substitute [tex]\( x \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 3\left(\frac{1}{4}\right) + 1 = \frac{3}{4} + 1 = \frac{7}{4} \][/tex]
Both lines intersect at [tex]\(\left(\frac{1}{4}, \frac{7}{4}\right)\)[/tex]. Therefore, the statement "The boundary lines intersect" is true.
### Summary
So, the true statements are:
- Both boundary lines are solid.
- A solution to the system is [tex]\((1, 3)\)[/tex].
- The boundary lines intersect.
Thus, the correct statements are:
- Both boundary lines are solid.
- A solution to the system is [tex]\((1, 3)\)[/tex].
- The boundary lines intersect.
### 1. "The slope of one boundary line is 2"
To find the slopes of the lines defined by the inequalities:
- For [tex]\( y \leq 3x + 1 \)[/tex]: The inequality is already in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Here, the slope [tex]\( m \)[/tex] is 3.
- For [tex]\( y \geq -x + 2 \)[/tex]: Similarly, this inequality is in slope-intercept form, and the slope [tex]\( m \)[/tex] is -1.
Hence, the slopes of the lines are 3 and -1. Neither of these slopes is 2. Therefore, the statement "The slope of one boundary line is 2" is false.
### 2. "Both boundary lines are solid"
Inequalities using [tex]\( \leq \)[/tex] (less than or equal to) and [tex]\( \geq \)[/tex] (greater than or equal to) dictate that the boundary lines are solid, indicating that the points on these lines are included in the solution set.
Therefore, the statement "Both boundary lines are solid" is true.
### 3. "A solution to the system is [tex]\((1, 3)\)[/tex]"
To check if the point [tex]\((1, 3)\)[/tex] is a solution to both inequalities:
- For [tex]\( y \leq 3x + 1 \)[/tex]: Plug in [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 \leq 3(1) + 1 \][/tex]
[tex]\[ 3 \leq 4 \][/tex]
This holds true.
- For [tex]\( y \geq -x + 2 \)[/tex]: Again, plug in [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 \geq -1 + 2 \][/tex]
[tex]\[ 3 \geq 1 \][/tex]
This also holds true.
Thus, the point [tex]\((1, 3)\)[/tex] satisfies both inequalities. Therefore, the statement "A solution to the system is [tex]\((1, 3)\)[/tex]" is true.
### 4. "Both inequalities are shaded below the boundary lines"
To determine where the areas are shaded relative to the boundary lines:
- For [tex]\( y \leq 3x + 1 \)[/tex]: The shading is below the line because [tex]\( y \)[/tex] is less than or equal to [tex]\( 3x + 1 \)[/tex].
- For [tex]\( y \geq -x + 2 \)[/tex]: The shading is above the line because [tex]\( y \)[/tex] is greater than or equal to [tex]\( -x + 2 \)[/tex].
As the shadings are in opposite directions for each inequality, the statement "Both inequalities are shaded below the boundary lines" is false.
### 5. "The boundary lines intersect"
To find if and where the boundary lines intersect, we set the equations equal to each other and solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ 3x + 1 = -x + 2 \][/tex]
Solving this:
[tex]\[ 3x + x = 2 - 1 \][/tex]
[tex]\[ 4x = 1 \][/tex]
[tex]\[ x = \frac{1}{4} \][/tex]
Substitute [tex]\( x \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 3\left(\frac{1}{4}\right) + 1 = \frac{3}{4} + 1 = \frac{7}{4} \][/tex]
Both lines intersect at [tex]\(\left(\frac{1}{4}, \frac{7}{4}\right)\)[/tex]. Therefore, the statement "The boundary lines intersect" is true.
### Summary
So, the true statements are:
- Both boundary lines are solid.
- A solution to the system is [tex]\((1, 3)\)[/tex].
- The boundary lines intersect.
Thus, the correct statements are:
- Both boundary lines are solid.
- A solution to the system is [tex]\((1, 3)\)[/tex].
- The boundary lines intersect.
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