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Magazine Chapter 7: Problem 7
Graph inequality. \(y>2 x-1\)
Short Answer
Expert verified
The graph of the inequality \(y > 2x - 1\) is the entire region above the dashed line \(y = 2x - 1\).
Step by step solution
01
Determine the Boundary Line
The inequality \(y > 2x - 1\) can be treated as an equation \(y = 2x - 1\) to find the boundary line. This equation is in the slope-intercept form \(y = mx + b\) where m is the slope and b is the y-intercept. Here, the slope (m) is 2 and the y-intercept (b) is -1. This means the line crosses the y-axis at -1 and the slope of 2 signifies that for every 2 units move up, one unit move to the right.
02
Graph the Boundary Line
Draw a line that starts at the point (0,-1) and has a slope of 2. Since the inequality is 'greater than' (>) and not 'greater than or equal to' (鈮�), the boundary line is dashed, not solid.
03
Choose a Test Point
Choose a point not on the boundary line to test the inequality. The point (0,0) is often a good choice if it is not on the boundary line.
04
Substitute the Test Point into the Inequality
Substitute (0,0) into the original inequality \(y > 2x - 1\). It becomes 0 > 2(0) - 1 which simplifies to 0 > -1.
05
Determine Which Half to Shade
Since the inequality 0 > -1 is true, the half of the plane that contains the test point (0,0) is the solution to the inequality. Hence, the area above the line \(y = 2x - 1\) is shaded.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form is a fundamental concept when dealing with linear equations. It is expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope \(m\) signifies the rate at which \(y\) changes with respect to \(x\).
In simpler terms, it tells you how steep the line is. For instance, a slope of 2 implies that the line rises by 2 units for every 1 unit it moves horizontally.
The y-intercept \(b\) is the point where the line crosses the y-axis. In the equation \(y = 2x - 1\), the slope is 2 and the y-intercept is -1.
This means our line starts at -1 on the y-axis and climbs up as it moves to the right.
In simpler terms, it tells you how steep the line is. For instance, a slope of 2 implies that the line rises by 2 units for every 1 unit it moves horizontally.
The y-intercept \(b\) is the point where the line crosses the y-axis. In the equation \(y = 2x - 1\), the slope is 2 and the y-intercept is -1.
This means our line starts at -1 on the y-axis and climbs up as it moves to the right.
Boundary Line
A boundary line in graphing inequalities is the line that separates the coordinate plane into two halves. It is drawn using the related equation from the inequality. For example, with the inequality \(y > 2x - 1\), the boundary line is \(y = 2x - 1\).
Whether the line is solid or dashed depends on the inequality:
Here, because we have a 'greater than' inequality (\(>\)), the boundary is a dashed line.
Whether the line is solid or dashed depends on the inequality:
- A solid line indicates the 'equal to' also applies (\(\geq\) or \(\leq\)).
- A dashed line shows that exact values on the line are not included (\(>\) or \(<\)).
Here, because we have a 'greater than' inequality (\(>\)), the boundary is a dashed line.
Test Point
A test point is used to decide which side of the boundary line to shade when graphing inequalities. Pick any point not on the boundary line鈥攐ften, (0,0) is a convenient choice.
If substituting this point into the original inequality results in a true statement, then that point is in the solution region, and that side of the boundary line should be shaded.
Let's say we use the point (0,0) for the inequality \(y > 2x - 1\). Substituting gives 0 > -1, which is true, indicating that the side of the line containing (0,0) is the solution region. So, the area above our line is shaded.
If substituting this point into the original inequality results in a true statement, then that point is in the solution region, and that side of the boundary line should be shaded.
Let's say we use the point (0,0) for the inequality \(y > 2x - 1\). Substituting gives 0 > -1, which is true, indicating that the side of the line containing (0,0) is the solution region. So, the area above our line is shaded.
Coordinate Plane
The coordinate plane is a two-dimensional plane formed by the intersection of the x-axis (horizontal) and the y-axis (vertical). It helps us graphically represent equations and inequalities.
Each point on this plane is identified by a pair of coordinates \((x, y)\).
When graphing inequalities, the coordinate plane helps us visualize the solution sets. By plotting the boundary line and using a test point, as explained, you can easily see which region of the plane represents the solutions to the inequality.
Each point on this plane is identified by a pair of coordinates \((x, y)\).
- The x-coordinate indicates the distance along the horizontal axis.
- The y-coordinate shows the distance along the vertical axis.
When graphing inequalities, the coordinate plane helps us visualize the solution sets. By plotting the boundary line and using a test point, as explained, you can easily see which region of the plane represents the solutions to the inequality.
