Question

Graph each inequality. $$x \geq 0$$

Short answer

Shade the region to the right of the y-axis and include the y-axis as a solid line.

Step-by-step solution

Understand the Inequality

The inequality given is \(x \geq 0\). This means that all values of x must be greater than or equal to 0.

Identify the Boundary Line

The boundary line for this inequality is at \(x = 0\). On a coordinate plane, this is the y-axis. Since the inequality includes \(=\), the line \(x = 0\) will be solid.

Determine the Shaded Region

Since the inequality is \(x \geq 0\), shade the entire region to the right of the y-axis, including the y-axis itself. This indicates that all points with \(x \geq 0\) satisfy the inequality.

Final Graph

Draw the y-axis and make it a solid line. Shade everything to the right of the y-axis to represent all x-values that are greater than or equal to zero.

Key concepts

coordinate plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It has two main axes: the x-axis and the y-axis. The x-axis runs horizontally, while the y-axis runs vertically. These two axes intersect at the origin, which is the point (0,0). The coordinate plane is divided into four quadrants, each representing positive or negative values for x and y. Understanding the coordinate plane is essential for graphing inequalities because it helps us identify regions that satisfy the given conditions.

boundary line

A boundary line is a crucial part of graphing inequalities. It represents the set of points where the inequality changes from true to false or vice versa. In our exercise, the boundary line is given by the equation \(x = 0\). Since the inequality is \(x \geq 0\), our boundary line is solid. A solid line means that points on the line are included in the solution. To graph this, we plot a vertical line along the y-axis, which serves as the boundary between the region where \(x \geq 0\) is true and where it is not.

shaded region

After plotting the boundary line, the next step is shading the region that satisfies the inequality. For \(x \geq 0\), we need to shade the entire area to the right of the y-axis. This shaded region includes the boundary line itself since the inequality is \(\geq\), not just \(>\). Shading helps visualize all the possible solutions that make the inequality true. Only areas within this shaded region meet the condition \(x \geq 0\).

inequality

The inequality \(x \geq 0\) states that x must be greater than or equal to zero. Inequalities like this one are different from equations because they include a range of solutions instead of just one. When graphing, understanding whether the inequality is strict (using \(<\) or \(>\)) or inclusive (using \(\leq\) or \(\geq\)) is important. For \(x \geq 0\), the boundary line is solid because every point on the line \(x=0\) satisfies the inequality. By identifying the boundary line and shading the appropriate region, we can graphically represent all the solutions that satisfy the inequality.