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Chapter 7: Problem 22

Graph each linear inequality. \(x \geq 0\)

Short Answer

Expert verified
The graph of the inequality \(x \geq 0\) is a shaded region to the right of the line \(x = 0\) (including the line itself), indicating all the \(x\) values that are greater than or equal to zero.

Step by step solution

01

Plotting the Line

Firstly, a line representing the equation \(x = 0\) should be drawn. This is a vertical line that intersects the y-axis at all points where \(x\) is 0.
02

Representing the Inequality

Next, the inequality is to be illustrated. As it is \(x \geq 0\), the region where \(x\) is greater than or equal to 0 should be shaded. In this scenario, that would be the area on the right side of the \(x = 0\) line. This includes the line \(x = 0\) as well since the inequality isn't just \(x > 0\), but \(x \geq 0\).
03

Final Touches

Lastly, make sure the origin (0,0) is clearly labelled on the graph and the graph should align with the inequality direction. All the area to the right of the line and the line itself should be shaded.

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Key Concepts

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

Linear Equations
Linear equations are equations of the first degree, meaning they have no exponents greater than one. They form straight lines when graphed on a coordinate plane.

These equations often appear in the form \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are constants. Lines produced by linear equations can be vertical, horizontal, or diagonal, depending on these constants.
  • A horizontal line, such as \(y = b\), means all points on the line have the same y-value.
  • A vertical line, like \(x = a\), signifies all points share the same x-value.
  • Diagonal lines are generated by equations like \(y = mx + c\), where \(m\) represents the slope and \(c\) the y-intercept.
Understanding linear equations is crucial for graphing inequalities, as these equations set the boundaries we work with.
Unequal Relations
Unlike equations, inequalities show a relationship where one side is not necessarily equal to the other. They use signs like \(>\), \(<\), \(\geq\), and \(\leq\) to indicate this.

In graphing contexts, inequalities define regions rather than lines. For example, in the inequality \(x \geq 0\), each point where \(x\) is greater than or equal to zero is part of the solution set.
  • \(>\) or \(<\) signifies that the boundary is not included, often depicted with a dashed line.
  • \(\geq\) or \(\leq\) implies the boundary is included, usually shown with a solid line.
By understanding inequalities, we can shade the regions on a graph accurately to denote where the inequality holds true.
Plotting Graphs
Plotting graphs involves translating equations or inequalities onto a visual plane to provide insight into their properties. This is particularly useful for inequalities since they represent areas on a graph.

When plotting linear inequalities like \(x \geq 0\), begin by drawing the boundary line鈥攈ere, a vertical line at \(x = 0\).
  • Use a solid line for \(\geq\) or \(\leq\) to show that the boundary is part of the solution.
  • Select and test points on either side of the line to determine which region to shade.
  • Shade the correct region to indicate which values satisfy the inequality.
Plotting helps visualize the solutions and provides a comprehensive understanding of the inequality at hand.
Coordinate Plane
The coordinate plane is a two-dimensional surface used to graphically represent equations and inequalities. It comprises two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).

Each point on this plane is defined by an \((x, y)\) coordinate. Here's a quick refresher on its basic characteristics:
  • The x-axis divides the plane into upper and lower sections.
  • The y-axis divides the plane into left and right sections.
  • Quadrants are created by the intersection of the axes, numbered I through IV, going counterclockwise starting from the top right.
In working with inequalities, knowing the coordinate plane allows you to accurately plot critical points and shaded regions, providing clarity on what solution sets look like in a visual format.

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Most popular questions from this chapter

Graph each linear inequality. \(2 y-x>4\)

In Exercises 23-38, graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}3 x+6 y \leq 6 \\ 2 x+y \leq 8\end{array}\right.\)

Use the directions for Exercises 9-14 to graph each quadratic function. Use the quadratic formula to find \(x\)-intercepts, rounded to the nearest tenth. \(f(x)=-3 x^{2}+6 x-2\)

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \boldsymbol{x}, \text { Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y}, \\ \text { Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 0.5 & 7.4 \\ \hline 1.5 & 9 \\ \hline 4 & 6 \\ \hline \end{array} $$ $$ \begin{array}{|l|} \hline u a d R e g \\ y=a \times 2+b x+c \\ a=-.8 \\ b=3.2 \\ c=6 \end{array} $$ a. Explain why a quadratic function was used to model the data. Why is the value of \(a\) negative? b. Use the graphing calculator screen to express the model in function notation. c. Use the model from part (b) to determine the \(x\)-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the ball occurs _____ feet from where it was thrown, and the maximum height is _____ feet.

Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by linear, exponential, logarithmic, or quadratic functions. Group members should select the two sets of data that are most interesting and relevant. Then consult a person who is familiar with graphing calculators to show you how to obtain a function that best fits each set of data. Once you have these functions, each group member should make one prediction based on one of the models, and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?
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