Problem 21 - Algebra 2

Question

Graph each inequality. $$ y > |4 x| $$

Step-by-Step Solution

Answer

Shade the area above the dashed 'V' shape formed by \(y = |4x|\).

Step 1: Understand the Absolute Value Graph

The function involves an absolute value expression, which means the graph will have a 'V' shape. For \( y = |4x| \), the vertex of the graph is at the origin \((0,0)\) and it opens upward.

Step 2: Graph the Line \( y = |4x| \)

Begin by sketching the graph of \( y = |4x| \). This involves drawing two lines: one with a slope of 4 and passing through the origin, and another with a slope of -4, also passing through the origin. This creates the 'V' shape.

Step 3: Determine the Inequality Region

Since the inequality is \( y > |4x| \), the solution region is above the graph of \( y = |4x| \). This means we shade the region that is above both arms of the 'V' shape.

Step 4: Use Dashed Line for Boundary

Because the inequality symbol is 'greater than' (\(>\)) and not 'greater than or equal to' (\(\geq\)), the boundary line (the graph of \( y = |4x| \)) should be dashed to indicate that points on the line are not included in the solution.

Key Concepts

Graphing Inequalities

Graphing inequalities involves finding the set of points on a coordinate plane that satisfy a given inequality. When dealing with absolute value inequalities like \( y > |4x| \), this means we are looking for the region above the "V" shaped graph created by the absolute value function. The process starts by graphing the related equation, in this case, \( y = |4x| \). Once we have this visual representation, the next step is to identify the area that satisfies the inequality, which involves shading the appropriate region.When shading, it's crucial to remember:

• The inequality symbol \(>\) means we are interested in values where \(y\) is strictly greater, rather than greater than or equal to.
• Carefully consider the direction of the "V" to determine which region to shade above or below.
• Comparator signs influence whether the boundary is part of the solution, affecting whether we shade inside or outside the boundary lines.

Coordinate Plane

A coordinate plane is a two-dimensional surface on which we can graph equations and inequalities. It has two number lines: one horizontal (the x-axis) and one vertical (the y-axis), meeting at a point called the origin, \(0,0\).The coordinate plane is essential for:

• Plotting precise points determined by coordinates.
• Visualizing graphs of equations and inequalities.
• Making it easier to understand and analyze relationships between variables.

For inequalities like \( y > |4x| \), we plot the boundary created by the related equation \( y = |4x| \). It helps us understand where the inequality "lives" by showing which side of the boundary we need to consider.

Shading Regions

Shading regions is an important step in solving and representing inequalities graphically. Once you identify the inequality's boundary, you must decide which side of the boundary the solution set lies. In the case of \( y > |4x| \), we need to shade the region above the "V" because the inequality states that \(y\) is greater than the absolute value expression.Steps to correctly shade regions include:

• Graph the boundary line using the equation derived from the inequality, in this case \( y = |4x| \).
• Decide which side of the line satisfies the inequality. Start by picking a test point not on the boundary, typically the origin \(0,0\) can be useful unless it lies on the boundary.
• If the test point satisfies the inequality, shade that side, otherwise shade the opposite side.

Understanding and correctly shading the right region is crucial because it visually represents all solutions to the inequality.

Dashed Boundary Lines

Dashed boundary lines are utilized in graphing to show that points on the line are not part of the solution set for an inequality. This is the case when the inequality is either \(>\) or \(<\). In our exercise with \( y > |4x| \), the boundary line is drawn as a dashed line to clearly indicate that the points exactly on the boundary \( y = |4x| \) are not included in our solution. When to use dashed lines:

• When the inequality symbol is "strict" (\(>\) or \(<\)).
• To ensure the graph accurately represents only the values strictly greater than (or less than) the boundary.

Using dashed lines effectively differentiates between values included in the inequality's solution and those that are not, ensuring clarity in visual representation.