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Chapter 6: Problem 54

Sketch the graph of the inequality. $$3 x-y \geq 5$$

Short Answer

Expert verified
The sketch of the graph of the inequality \(3x - y \geq 5\) will be the line \(y = 3x - 5\), with the region above the line shaded, as that is the region which satisfies the inequality.

Step by step solution

01

Converting Inequality to Equation

Convert the given inequality \(3x - y \geq 5\) to an equation by replacing the inequality sign '\(\geq\)' with an '=' sign. The new equation will be \(3x - y = 5\)
02

Draw the Graph for the Equation

To draw the graph, the equation can be re-arranged as \(y = 3x - 5\). This is the equation of a line with a y-intercept of -5 and a slope of 3. Plot this line on the graph.
03

Shading the Region

Now, one needs to shade the region of the graph that satisfies the inequality. Because the inequality is '\(\geq\)', the region to be shaded will be above the line for \(y=3x-5\). This is the valid area for the given inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality to Equation Conversion
Understanding how to convert a linear inequality to an equation is a fundamental step in graphing and solving inequalities. When working with an inequality such as \(3x - y \geq 5\), the initial goal is to represent this relationship as an equation. This is accomplished by changing the inequality sign to an equal sign, resulting in \(3x - y = 5\).

This creates a 'boundary' line that will be crucial in the graphing process. The original inequality indicates that all solutions lie on one side of this boundary. When converting, remember that the inequality expresses a range of values, whereas the equation represents a precise relationship between \(x\) and \(y\).

  • Boundary Line: The graph of \(3x - y = 5\) is a straight line dividing the plane into two halves.
  • Inequality Significance: The inequality sign tells us which side of the boundary line contains the solution set.
Slope-Intercept Form
The slope-intercept form of a line, given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, is crucial for graphing lines efficiently. For the equation \(3x - y = 5\), we can rearrange it to the slope-intercept form to get \(y = 3x - 5\).

The slope, \(3\), tells us how steep the line is and in which direction it goes. With every unit increase in \(x\), \(y\) increases by 3 units. The y-intercept, \(-5\), indicates where the line crosses the y-axis. This conversion to slope-intercept form provides us with key points that serve as a starting point for drawing the line on a graph.

  • Slope (\(m\)): Represents the line's rise over run. Positive slope goes upwards, while a negative slope goes downwards as \(x\) increases.
  • Y-intercept (\(b\)): The point where the line crosses the y-axis, which can be plotted directly onto the graph.
Shading Regions on Graphs
Shading indicates which side of the boundary line contains the solutions to the inequality. In our example, the inequality \(3x - y \geq 5\) suggests that valid points lie on the line or above it since 'greater than or equal to' includes the boundary line itself.

To shade correctly, a test point not on the line (like (0,0)) can be used. If the point satisfies the original inequality, the side containing the test point is shaded. Otherwise, the opposite side is. Here, since \(3(0) - (0) \geq 5\) is not true, we shade the opposite side of the line from the origin.

  • Test Point: Helps determine the correct region to shade. If the inequality holds for the test point, shade that side; if not, shade the other.
  • Graphical Representation: The shaded region visually represents all possible solutions to the inequality.

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Most popular questions from this chapter

Is each ordered pair a solution of the inequality? $$4 x+7 y \leq 26 ;(3,2),(2,3)$$

Find the slope and the y-intercept of the line. $$y=-5 x+2$$

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