Question

Sketch the graph of the inequality. $$x \geq-9$$

Short answer

The graph of the inequality \(x \geq -9\) is a number line where the interval \([-9, +\infty)\) is shaded.

Step-by-step solution

Understand the inequality

The inequality is \(x \geq -9\). Here, x is greater than or equal to -9.

Define the interval

The values of x that satisfy this inequality form an interval of numbers. In this case, the interval is \([-9, +\infty)\). This means that all real numbers from -9 to positive infinity, including -9, are the solution of the inequality.

Draw the number line

Draw a number line and plot the interval on this line. The number -9 will be included in the interval, so it is represented by a closed point. The number line extends to positive infinity, which is represented by an arrowhead.

Shade the area

The points on the number line that represent the solution of the inequality are shaded. In this case, all points on the line to the right of -9 (inclusive) are the solution, so they are shaded.

Key concepts

Number Line Representation

Graphical representation of numerical inequalities on a number line offers a clear visual understanding of which values satisfy the inequality. For instance, when asked to sketch the graph of the inequality \(x \geq -9\), it's essential to translate the inequality into a visual format.

To achieve this, one begins with a horizontal line, which symbolizes the number line; a collection of points where each point corresponds to a real number. The position of each number on this line is fixed: numbers increase as you move right and decrease as you move left. For the given inequality, the crucial point of reference is -9. It's on this number that one will anchor the visual representation of all values that make the inequality true.

Inequality Solutions

The solutions to inequalities are not just single numbers, but rather ranges or intervals of numbers that satisfy the inequality. Considering the inequality \(x \geq -9\), it implies that the solution includes not only -9 but also every number greater than -9.

This set of numbers is called an interval, and it's expressed as \([-9, +\infty)\), to show that the solution starts at -9 and extends without bound toward positive infinity. When graphing this on a number line, a visual stretch from -9 onward to the right depicts this unending range. To convey the complete solution, these numbers are typically shaded or marked along the number line.

Closed Point Notation

The details of interval notation often translate into specific symbols on a number line. For instance, in the inequality \(x \geq -9\), the use of \(\geq\) indicates that -9 is as much a solution as any number greater than it. As such, on the number line, -9 is designated using a closed point (often a filled circle) to signify its inclusion in the solution set.

The closed point is a pivotal element of number line diagrams that highlights the inclusion of an endpoint of an interval. Unlike an open point (an unfilled circle), which would signify that the endpoint is not part of the solution, the closed point demarcates inclusiveness clearly, ensuring there's no ambiguity regarding which numbers satisfy the inequality. This distinction is fundamental to correctly interpreting and graphing inequalities on the number line.