Problem 33 - Intermediate Algebra

Question

Graph each inequality in two variables. $$ x>-3 $$

Step-by-Step Solution

Answer

Graph the vertical dashed line at \(x = -3\) and shade the region to the right.

Step 1: Identify the Inequality

The inequality given is \(x > -3\). This inequality means that the solution includes all values of \(x\) that are greater than \(-3\). It is a vertical line boundary on the graph.

Step 2: Draw the Boundary Line

Plot the vertical line \(x = -3\) on the coordinate plane. Since the inequality is strictly greater than \(-3\), the line \(x = -3\) should be drawn as a dashed line to indicate that values on the line itself are not included in the solution.

Step 3: Shade the Solution Region

Shade the region to the right of the line \(x = -3\), as this area represents all the points where \(x\) is greater than \(-3\). Make sure the shading does not include the line itself.

Key Concepts

Two-variable inequalities

When dealing with inequalities involving two variables, it's important to understand how they differ from simple equations. Unlike an equation that shows an exact relationship between variables, an inequality shows a range of values. This exercise involves the inequality \(x > -3\).

• Two-variable inequalities involve expressions with two different variables, usually \(x\) and \(y\).
• They represent a broad set of solutions rather than a single line, meaning they define a region on a graph.
• The solution set includes all points that satisfy the inequality.

Graphing these inequalities helps visually interpret which areas on the graph meet the inequality's conditions. This visual assistance can often make it easier to understand and solve problems.

Coordinate Plane

The coordinate plane is the stage where you graph two-variable inequalities. It consists of two perpendicular axes: the horizontal axis (commonly referred to as the \(x\)-axis) and the vertical axis (known as the \(y\)-axis). Each point on this plane is defined by a pair of numerical values, \( (x, y) \).

• Every point on the plane corresponds to an \(x\) and \(y\) value.
• It is crucial for graphing as it provides a visual representation of mathematical relationships.
• In the case of a single-variable like \(x > -3\), you will consider one axis predominantly.

Knowing how to navigate the coordinate plane is essential for graphing inequalities accurately. It is the first step in visualizing the areas where these inequalities hold true.

Solution Region

The solution region is the area on the coordinate plane where all the solutions to an inequality lie. For the inequality \(x > -3\), the solution region includes all the points to the right of the boundary line \(x = -3\).

• This region is where all conditions of the inequality are met.
• It is often represented by shading the appropriate side of a boundary line on the graph.
• In "strict" inequalities like \(x > -3\), the boundary line itself is not part of the solution.

To identify the correct region, plot the boundary line first and then shade the region that satisfies the inequality condition. This shading helps visualize which part of the graph meets the inequality's requirements.

Boundary Lines

Boundary lines like \(x = -3\) in inequalities act as borders for the solution region. They play a crucial role in graphing because they help define where the inequality starts or stops.

• A solid line is used for \(\leq\) or \(\geq\) inequalities, indicating that points on the line are part of the solution.
• A dashed line, like the one for \(x = -3\), signifies a strict inequality such as \(x > -3\), indicating the line itself is not included in the solution.
• Drawing the correct type of line ensures an accurate representation of the inequality.

Understanding how to use and interpret boundary lines is vital for accurately solving and graphing inequalities. This step helps you visualize and comprehend the precise limits of an inequality's solution set.