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To analyze the statements about the graph of the inequalities [tex]\( y \leq 3x + 1 \)[/tex] and [tex]\( y \geq -x + 2 \)[/tex], let's go through each statement step-by-step.
1. The slope of one boundary line is 2:
- The first inequality is [tex]\( y \leq 3x + 1 \)[/tex], which has a boundary line [tex]\( y = 3x + 1 \)[/tex] with a slope of 3.
- The second inequality is [tex]\( y \geq -x + 2 \)[/tex], which has a boundary line [tex]\( y = -x + 2 \)[/tex] with a slope of -1.
- Since neither slope is 2, this statement is false.
2. Both boundary lines are solid:
- The inequality [tex]\( y \leq 3x + 1 \)[/tex] uses [tex]\( \leq \)[/tex], which means the boundary line [tex]\( y = 3x + 1 \)[/tex] is solid.
- The inequality [tex]\( y \geq -x + 2 \)[/tex] uses [tex]\( \geq \)[/tex], which means the boundary line [tex]\( y = -x + 2 \)[/tex] is solid.
- Therefore, this statement is true.
3. A solution to the system is [tex]\((1, 3)\)[/tex]:
- Substituting [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex] into the first inequality [tex]\( y \leq 3x + 1 \)[/tex]:
[tex]\[ 3 \leq 3 \cdot 1 + 1 \implies 3 \leq 4 \quad \text{(true)} \][/tex]
- Substituting [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex] into the second inequality [tex]\( y \geq -x + 2 \)[/tex]:
[tex]\[ 3 \geq -1 + 2 \implies 3 \geq 1 \quad \text{(true)} \][/tex]
- Since [tex]\((1, 3)\)[/tex] satisfies both inequalities, this statement is true.
4. Both inequalities are shaded below the boundary lines:
- The inequality [tex]\( y \leq 3x + 1 \)[/tex] is shaded below the line [tex]\( y = 3x + 1 \)[/tex].
- The inequality [tex]\( y \geq -x + 2 \)[/tex] is shaded above the line [tex]\( y = -x + 2 \)[/tex].
- Since the second inequality is not shaded below its boundary line, this statement is false.
5. The boundary lines intersect:
- To find the intersection point of the lines [tex]\( y = 3x + 1 \)[/tex] and [tex]\( y = -x + 2 \)[/tex], set the equations equal to each other:
[tex]\[ 3x + 1 = -x + 2 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 3x + x = 2 - 1 \implies 4x = 1 \implies x = \frac{1}{4} \][/tex]
Substituting [tex]\( x = \frac{1}{4} \)[/tex] into [tex]\( y = 3x + 1 \)[/tex]:
[tex]\[ y = 3 \cdot \frac{1}{4} + 1 = \frac{3}{4} + 1 = 1.75 \][/tex]
- Thus, the intersection point is [tex]\( \left( 0.25, 1.75 \right) \)[/tex], confirming the lines intersect. Therefore, this statement is true.
Summary:
- The slope of one boundary line is 2: False
- Both boundary lines are solid: True
- A solution to the system is [tex]\((1, 3)\)[/tex]: True
- Both inequalities are shaded below the boundary lines: False
- The boundary lines intersect: True
1. The slope of one boundary line is 2:
- The first inequality is [tex]\( y \leq 3x + 1 \)[/tex], which has a boundary line [tex]\( y = 3x + 1 \)[/tex] with a slope of 3.
- The second inequality is [tex]\( y \geq -x + 2 \)[/tex], which has a boundary line [tex]\( y = -x + 2 \)[/tex] with a slope of -1.
- Since neither slope is 2, this statement is false.
2. Both boundary lines are solid:
- The inequality [tex]\( y \leq 3x + 1 \)[/tex] uses [tex]\( \leq \)[/tex], which means the boundary line [tex]\( y = 3x + 1 \)[/tex] is solid.
- The inequality [tex]\( y \geq -x + 2 \)[/tex] uses [tex]\( \geq \)[/tex], which means the boundary line [tex]\( y = -x + 2 \)[/tex] is solid.
- Therefore, this statement is true.
3. A solution to the system is [tex]\((1, 3)\)[/tex]:
- Substituting [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex] into the first inequality [tex]\( y \leq 3x + 1 \)[/tex]:
[tex]\[ 3 \leq 3 \cdot 1 + 1 \implies 3 \leq 4 \quad \text{(true)} \][/tex]
- Substituting [tex]\( x = 1 \)[/tex] and [tex]\( y = 3 \)[/tex] into the second inequality [tex]\( y \geq -x + 2 \)[/tex]:
[tex]\[ 3 \geq -1 + 2 \implies 3 \geq 1 \quad \text{(true)} \][/tex]
- Since [tex]\((1, 3)\)[/tex] satisfies both inequalities, this statement is true.
4. Both inequalities are shaded below the boundary lines:
- The inequality [tex]\( y \leq 3x + 1 \)[/tex] is shaded below the line [tex]\( y = 3x + 1 \)[/tex].
- The inequality [tex]\( y \geq -x + 2 \)[/tex] is shaded above the line [tex]\( y = -x + 2 \)[/tex].
- Since the second inequality is not shaded below its boundary line, this statement is false.
5. The boundary lines intersect:
- To find the intersection point of the lines [tex]\( y = 3x + 1 \)[/tex] and [tex]\( y = -x + 2 \)[/tex], set the equations equal to each other:
[tex]\[ 3x + 1 = -x + 2 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 3x + x = 2 - 1 \implies 4x = 1 \implies x = \frac{1}{4} \][/tex]
Substituting [tex]\( x = \frac{1}{4} \)[/tex] into [tex]\( y = 3x + 1 \)[/tex]:
[tex]\[ y = 3 \cdot \frac{1}{4} + 1 = \frac{3}{4} + 1 = 1.75 \][/tex]
- Thus, the intersection point is [tex]\( \left( 0.25, 1.75 \right) \)[/tex], confirming the lines intersect. Therefore, this statement is true.
Summary:
- The slope of one boundary line is 2: False
- Both boundary lines are solid: True
- A solution to the system is [tex]\((1, 3)\)[/tex]: True
- Both inequalities are shaded below the boundary lines: False
- The boundary lines intersect: True
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