The graph of the inequality x > 4y + 1 is 1 point of the line x = 4y + 1. *
A. above
B. below
C. to the right
D. to the left
The Correct Answer and Explanation is :
The correct answer is C. to the right.
To understand this, let’s analyze the inequality ( x > 4y + 1 ) and its corresponding equation ( x = 4y + 1 ).
1. Graphing the Equation ( x = 4y + 1 ):
This equation represents a straight line. To express it in slope-intercept form (( y = mx + b )), solve for ( y ):
[
\begin{align} x &= 4y + 1 \ x – 1 &= 4y \ y &= \frac{1}{4}x – \frac{1}{4} \end{align}
]
Here, the slope (( m )) is ( \frac{1}{4} ), and the y-intercept (( b )) is ( -\frac{1}{4} ). This means the line crosses the y-axis at ( -\frac{1}{4} ) and rises ( 1 ) unit for every ( 4 ) units it runs to the right.
2. Interpreting the Inequality ( x > 4y + 1 ):
The inequality ( x > 4y + 1 ) indicates the region where ( x ) is greater than ( 4y + 1 ). Geometrically, this represents the area to the right of the line ( x = 4y + 1 ). To see why, consider any point ( (x, y) ) on the line: it satisfies ( x = 4y + 1 ). If we increase ( x ) while keeping ( y ) constant, the point moves to the right, and ( x ) becomes greater than ( 4y + 1 ). Conversely, decreasing ( x ) moves the point to the left, where ( x < 4y + 1 ).
3. Visual Confirmation:
To visualize this, plot the line ( x = 4y + 1 ). The region representing ( x > 4y + 1 ) lies entirely to the right of this line. Any point in this region will have an ( x )-coordinate greater than ( 4y + 1 ) for its corresponding ( y )-coordinate.
In summary, the inequality ( x > 4y + 1 ) defines the set of points located to the right of the line ( x = 4y + 1 ). Therefore, the correct answer is C. to the right.
For a visual demonstration of graphing a similar equation, you can refer to the following video: