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Magazine Chapter 2: Problem 15
Graph the solution set. $$y<2 x-1$$
Short Answer
Expert verified
Graph the line \( y = 2x - 1 \) as dashed, and shade the area below it.
Step by step solution
01
Understand the Inequality
The inequality given is \( y < 2x - 1 \). This represents a region below the line \( y = 2x - 1 \) in the coordinate plane. We need to graph this region.
02
Graph the Boundary Line
First, graph the boundary line \( y = 2x - 1 \). This line is dashed because the inequality is strict (\(<\) rather than \(\leq\)), indicating points on the line are not included in the solution set.
03
Identify the Slope and Y-intercept
The equation \( y = 2x - 1 \) is in slope-intercept form \( y = mx + b \) where \( m = 2 \) (slope) and \( b = -1 \) (y-intercept). Begin by plotting the y-intercept (0,-1) on the graph.
04
Plot Another Point Using the Slope
From the y-intercept (0, -1), use the slope \(m = 2\) which is \(\frac{2}{1}\), meaning 'rise' 2 units and 'run' 1 unit to plot a second point (1, 1) on the graph.
05
Draw the Dashed Line
Connect the points (0, -1) and (1, 1) with a dashed line to represent \( y = 2x - 1 \). This line is the boundary of the region where the inequality holds.
06
Shade the Correct Region
Since the inequality is \( y < 2x - 1 \), shade the region below the dashed line. This region represents all coordinates \((x, y)\) that satisfy the inequality.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope
The slope of a line is a key concept when graphing inequalities. The slope of a line determines how steep the line is.
When we look at the equation of a line in the slope-intercept form:
This means for every unit you move horizontally to the right along the x-axis, the line moves up 2 units vertically.
The slope is a ratio of 'rise' over 'run'. So, a slope of 2 can be thought of as
When we look at the equation of a line in the slope-intercept form:
- \( y = mx + b \)
- \( m \) stands for the slope.
This means for every unit you move horizontally to the right along the x-axis, the line moves up 2 units vertically.
The slope is a ratio of 'rise' over 'run'. So, a slope of 2 can be thought of as
- a rise of 2 units
- and a run of 1 unit.
Y-Intercept
The y-intercept is another important component when graphing inequalities. It is the point where the line crosses the y-axis.
In the slope-intercept form
This means the line crosses the y-axis at the point (0, -1).
To begin plotting our inequality, we start at this y-intercept point on the graph.
The y-intercept gives us a starting point for the line, making it easier to visualize where our line will be located on the coordinate plane.
Once you have the y-intercept, you can use the slope to plot other points, helping to draw the complete line on the graph.
In the slope-intercept form
- \( y = mx + b \)
- \( b \) represents the y-intercept.
This means the line crosses the y-axis at the point (0, -1).
To begin plotting our inequality, we start at this y-intercept point on the graph.
The y-intercept gives us a starting point for the line, making it easier to visualize where our line will be located on the coordinate plane.
Once you have the y-intercept, you can use the slope to plot other points, helping to draw the complete line on the graph.
Coordinate Plane
The coordinate plane is a two-dimensional space where each point is represented by a pair of numbers (x,y).
It consists of :
When graphing the inequality \( y < 2x - 1 \), the entire process takes place on this coordinate plane:
It consists of :
- a horizontal axis, known as the x-axis
- and a vertical axis, the y-axis.
When graphing the inequality \( y < 2x - 1 \), the entire process takes place on this coordinate plane:
- First, you identify the y-intercept,
- then use the slope to plot another point,
- and finally draw the boundary line.
Dashed Line
In graphing inequalities, the type of line used is significant.
In our exercise, we used a dashed line.
The dashed line is used when the inequality does not include the points on the line itself, which is when you have symbols like \(<\) or \(>\), as opposed to \(\leq\) or \(\geq\).
In \( y < 2x - 1 \), the inequality symbol \(<\) indicates that points on the line do not satisfy the inequality.
In our exercise, we used a dashed line.
The dashed line is used when the inequality does not include the points on the line itself, which is when you have symbols like \(<\) or \(>\), as opposed to \(\leq\) or \(\geq\).
In \( y < 2x - 1 \), the inequality symbol \(<\) indicates that points on the line do not satisfy the inequality.
- The dashed line represents these boundary points, showing that while they are not part of the solution set.
- Only the region below this line is shaded, indicating the actual solutions.
