Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x>y^{2} \\ x

Step-by-Step Solution

Answer

The solution to the system of inequalities is the overlap between the individual solutions for each inequality. In this case, it's the area to the right of the parabola \(x=y^{2}\) and below the line \(x=y+2\).

Step 1: Graph the First Inequality

Start by graphing the inequality \(x>y^{2}\). This is a parabola opening to the right. Draw a dashed line because the original inequality does not include the values of (x,y) that lie on the boundary line \(x=y^{2}\). The region to the right (greater than) of this dashed line will be shaded.

Step 2: Graph the Second Inequality

Next, graph the inequality \(x

Step 3: Determine the Solution Region

After both inequalities are plotted, determine the region where the shaded areas of both inequalities overlap - this represents the solution to the system. This region satisfies both inequalities simultaneously and represents the set of all possible solutions to the system.

Key Concepts

Graphing Inequalities

When graphing inequalities, we represent solutions on a coordinate plane, which allows us to visualize the range of possible answers. Unlike equations, which show equality, inequalities show a relationship of less than, greater than, less than or equal to, or greater than or equal to. For instance, when you come across the inequality \( x > y^2 \), you are actually looking at a set of points \( x \), which are greater than the square of \( y \). This creates a region in the graph where all the solutions lie.

In graphing such inequalities, consider the following steps: First, graph the boundary, which would be the equation you'd get if you replaced the inequality with an equals sign. For \( x > y^2 \), graph the parabola \( x = y^2 \). Next, decide whether to use a solid or dashed line. Use a solid line if the inequality includes the boundary (corresponding to \( \leq \) or \( \geq \)) and a dashed line if it does not (\( < \) or \( > \)). Lastly, shade the region that satisfies the inequality. For \( x > y^2 \), you shade to the right of the parabola.

Parabolas in Algebra

Parabolas are U-shaped curves that can open up, down, left, or right, depending on the quadratic function's form. In the context of graphing inequalities, when we see something like \( x > y^2 \), we're dealing with a horizontal parabola that opens to the right. The vertex of this parabola is at the origin (0,0).

Understanding how to draw parabolas is crucial in algebra and comes in handy when plotting the solutions for quadratic inequalities. To graph a parabola, one typically starts by identifying the vertex and then plotting points around it, ensuring symmetry. The direction of opening of the parabola indicates the side where the solution region will be, so it's essential to pay attention to it. In our example, since the parabola opens to the right, the solutions for the inequality lie to the right of it.

Quadratic Inequalities

Quadratic inequalities, like the one in our exercise \( x > y^2 \), involve a parabola. They differ from linear inequalities because the solutions form a two-dimensional area on the graph rather than a single line.

To graph a quadratic inequality, begin by graphing the corresponding parabola, and then establish which side of the parabola is the solution area. It can be inside the parabola for inequalities like \( x \leq y^2 \) or outside for ones like \( x \geq y^2 \). The inequality's symbol usually determines this. In our case, since it's \( x > y^2 \), the solution region is outside the parabola and to its right because 'greater than' refers to the values of \( x \) that are away from the vertex on the positive x-axis.

Linear Inequalities

Linear inequalities, such as \( x < y + 2 \), describe a relationship where one variable is greater than or less than the other by a fixed amount. To graph a linear inequality, start by graphing the corresponding equation as if it were an equality. In this example, \( x = y + 2 \) is a straight line with a slope of 1 and a y-intercept of 2.

Once you graph the boundary line for the inequality, choose a solid or dashed line based on whether the inequality includes the points on the line. Then shade above or below the line to indicate which side represents the solution set. For \( x < y + 2 \), the shading would be below the line since 'less than' points to the values of \( x \) that fall beneath the line.

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Problem 37

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Key Concepts

Graphing Inequalities

When graphing inequalities, we represent solutions on a coordinate plane, which allows us to visualize the range of possible answers. Unlike equations, which show equality, inequalities show a relationship of less than, greater than, less than or equal to, or greater than or equal to. For instance, when you come across the inequality \( x > y^2 \), you are actually looking at a set of points \( x \), which are greater than the square of \( y \). This creates a region in the graph where all the solutions lie.

In graphing such inequalities, consider the following steps: First, graph the boundary, which would be the equation you'd get if you replaced the inequality with an equals sign. For \( x > y^2 \), graph the parabola \( x = y^2 \). Next, decide whether to use a solid or dashed line. Use a solid line if the inequality includes the boundary (corresponding to \( \leq \) or \( \geq \)) and a dashed line if it does not (\( < \) or \( > \)). Lastly, shade the region that satisfies the inequality. For \( x > y^2 \), you shade to the right of the parabola.

Parabolas in Algebra

Parabolas are U-shaped curves that can open up, down, left, or right, depending on the quadratic function's form. In the context of graphing inequalities, when we see something like \( x > y^2 \), we're dealing with a horizontal parabola that opens to the right. The vertex of this parabola is at the origin (0,0).

Understanding how to draw parabolas is crucial in algebra and comes in handy when plotting the solutions for quadratic inequalities. To graph a parabola, one typically starts by identifying the vertex and then plotting points around it, ensuring symmetry. The direction of opening of the parabola indicates the side where the solution region will be, so it's essential to pay attention to it. In our example, since the parabola opens to the right, the solutions for the inequality lie to the right of it.

Quadratic Inequalities

Quadratic inequalities, like the one in our exercise \( x > y^2 \), involve a parabola. They differ from linear inequalities because the solutions form a two-dimensional area on the graph rather than a single line.

To graph a quadratic inequality, begin by graphing the corresponding parabola, and then establish which side of the parabola is the solution area. It can be inside the parabola for inequalities like \( x \leq y^2 \) or outside for ones like \( x \geq y^2 \). The inequality's symbol usually determines this. In our case, since it's \( x > y^2 \), the solution region is outside the parabola and to its right because 'greater than' refers to the values of \( x \) that are away from the vertex on the positive x-axis.

Linear Inequalities

Linear inequalities, such as \( x < y + 2 \), describe a relationship where one variable is greater than or less than the other by a fixed amount. To graph a linear inequality, start by graphing the corresponding equation as if it were an equality. In this example, \( x = y + 2 \) is a straight line with a slope of 1 and a y-intercept of 2.

Once you graph the boundary line for the inequality, choose a solid or dashed line based on whether the inequality includes the points on the line. Then shade above or below the line to indicate which side represents the solution set. For \( x < y + 2 \), the shading would be below the line since 'less than' points to the values of \( x \) that fall beneath the line.