Question

Graph the inequality: \(x \geq-2\)

Short answer

Draw a number line, plot a solid dot at \(-2\), and shade to the right.

Step-by-step solution

Understand the Inequality

First, recognize that the inequality given is \(x \geq -2\). This means that the solution includes all values of \(x\) that are greater than or equal to \(-2\).

Draw the Number Line

Draw a horizontal number line on your graph. Make sure to mark the point \(-2\) clearly on the number line.

Plot the Boundary Point

Since the inequality is greater than or equal to \(-2\), place a solid dot at the point \(-2\) on the number line. A solid dot signifies that \(-2\) is included in the solution.

Shade the Solution Region

Shade the line to the right of \(-2\) to indicate all the values that satisfy \(x \geq -2\). This shows that all numbers greater than \(-2\) are included in the solution.

Key concepts

number line

A number line is an essential tool for visualizing and solving inequalities. It is a straight, horizontal line with numbers placed at equal intervals along its length. When graphing an inequality, the number line helps you see which numbers satisfy the inequality.

In the context of the inequality \( x \geq -2 \), you start by drawing a horizontal line and marking the point \(-2\) on this line. The number line should extend to the left and right of \(-2\), allowing you to see both smaller and larger numbers in context.

The number line also tells us about the relationship between numbers. For example, any number to the right of \(-2\) indicates a greater value, while any number to the left signifies a smaller value. This visual aid helps you easily identify which numbers are solutions to the inequality.

inequality notation

Inequality notation is used to describe a range of values that satisfy a condition. In this exercise, the inequality is expressed as \( x \geq -2 \). Here’s a breakdown of what each part of this notation means:

• \( x \) : The variable we are working with.
• \( \geq \) : This symbol means 'greater than or equal to'.
• \( -2 \) : The boundary point in the inequality.

Understanding inequality notation helps in graphing because it defines the values we need to include. For instance, in \( x \geq -2 \), the inequality tells us that \(-2\) is part of the solution (hence, a solid dot) and that all numbers greater than \(-2\) (to the right on the number line) are also solutions.

Properly interpreting inequality notation ensures that we represent the problem accurately.

solution region

The solution region is the part of the number line that includes all values satisfying the inequality. In our example \( x \geq -2 \), the solution region begins at \(-2\) and extends indefinitely to the right.

To indicate this on the number line, place a solid dot at \(-2\) to show that \(-2\) is included in the solution set. Then, shade the region to the right of \(-2\). This shading visually represents all numbers greater than or equal to \(-2\).

By marking the solution region, you provide a clear visual reference that helps you and others quickly understand which values fulfill the inequality conditions. This clarity is especially useful when working with more complex inequalities or when comparing multiple inequalities.