Problem 12 - College Algebra with Modeling and Visualization

Question

Graph the solution set to the inequality. $$ 2 x^{2}-y<1 $$

Step-by-Step Solution

Answer

Graph \( y = 2x^2 - 1 \) as a dashed parabola and shade above it.

Step 1: Rewrite the Inequality

First, rewrite the inequality in a form that may be helpful for graphing. Given the inequality \( 2x^2 - y < 1 \), we can rearrange it to \( y > 2x^2 - 1 \). This will make it easier to consider the graph of the inequality.

Step 2: Identify the Boundary

The expression \( y = 2x^2 - 1 \) represents the boundary of the inequality \( y > 2x^2 - 1 \). This is a parabola that opens upwards, with its vertex at the point (0, -1). Determine the boundary by graphing this parabola as a dashed line because the inequality is strictly greater than (\( > \)), not greater than or equal to.

Step 3: Determine the Test Point

Choose a test point that is not on the boundary to determine which region to shade. A simple choice is the origin, \((0, 0)\). Substitute \(x = 0\) and \(y = 0\) into the inequality: \(0 > 2(0)^2 -1\) simplifies to \(0 > -1\), which is true.

Step 4: Graph and Shade the Correct Region

Since the test point satisfies the inequality, shade the region that contains the point \((0, 0)\). This region represents the set of points \((x, y)\) such that \(y > 2x^2 - 1\). Use a dashed line for the parabola to indicate that points on the line are not included in the solution set.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing solutions to inequalities on a coordinate plane. To graph an inequality such as \(y > 2x^2 - 1\), follow these steps:

• First, convert the inequality into an equation by changing the inequality symbol to an equal sign. For example, \(y = 2x^2 - 1\) creates the boundary of the inequality.
• Next, graph the boundary. Since the inequality is strict (represented by \( > \)), use a dashed line to indicate that points on the line are not part of the solution set.
• Then, choose a test point that is not on the line to determine which side of the boundary the solution set occurs. In this case, the test point \((0,0)\) satisfies the inequality, as substituting gives a true statement \(0 > -1\).
• Finally, shade the region where the inequality holds true. This shaded area represents all solutions (\(x, y\)) that satisfy the inequality.

Highlighting the dashed line for strict inequalities and ensuring correct shading are essential steps in graphing inequalities effectively.

Quadratic Inequalities

Quadratic inequalities involve expressions where the highest power of the variable is two. They can be tricky because their solutions are often ranges or regions. Consider the inequality \(2x^2 - y < 1\):

• Convert to a more common form by isolating \(y\), resulting in \(y > 2x^2 - 1\). Here, you are dealing with the inequality of a quadratic function.
• The inequality involves a parabola: \(y = 2x^2 - 1\), which has a vertex at (0, -1) and opens upwards.
• In quadratic inequalities, solving involves finding the regions that satisfy the inequality, frequently using a graphing technique as described in the previous section.

Understanding the shape and direction of the parabola helps when dealing with quadratic inequalities, as this impacts the region that represents the solution.

Inequality Solutions

Inequality solutions represent all the values that satisfy an inequality condition, such as within a particular area on a graph. For the given inequality \(y > 2x^2 - 1\):

• The solution is identified by the region in the graph where the inequality holds true. In this example, the solution is all points above the parabola.
• When you graph the inequality, remember to test points to ensure you are shading the correct region. This step is crucial because it verifies that you are identifying the solved area correctly.
• Solutions to inequalities can be displayed visually as well as written using interval notation if applicable, indicating the ranges of \(x\) or \(y\) that meet the inequality criteria.

Clearly defining boundaries and solutions helps in understanding and solving inequalities confidently.