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Magazine Chapter 9: Problem 6
\( y^{2}-x \leq 0\)
Short Answer
Expert verified
The solution is the region where \( y^2 \leq x \), including the parabola \( y^2 = x \).
Step by step solution
01
Rewrite the Inequality
The given inequality is \( y^2 - x \leq 0 \). To better understand it, we can rewrite it in the form \( y^2 \leq x \). This tells us that the value of \( y^2 \) should be less than or equal to \( x \).
02
Determine the Set of Solutions
Since we need \( y^2 \leq x \), we are looking for all points \((x, y)\) such that the square of \( y \) is less than or equal to \( x \). This represents the region including and below the parabola \( y^2 = x \) in the coordinate plane.- For any point on the parabola, \( y^2 = x \), which satisfies the inequality.- For any point below the parabola, \( y^2 < x \), which also satisfies the inequality.
03
Analyze the Boundary and Inside
The parabola \( y^2 = x \) is the boundary of the solutions. Points on the parabola make the inequality true because \( y^2 = x \) implies \( y^2 \leq x \). Points inside the region (where \( y^2 < x \)) also satisfy the inequality. Therefore, the solution includes all points \((x, y)\) where \( y^2 \leq x \).
04
Describe the Solution Set
The solution set is the entire region on the coordinate plane starting from any \( x \)-value greater than or equal to 0, and for each such \( x \), choosing a \( y \) value such that \( y^2 \leq x \). Visually, it is the area inside and on the right part of the parabola opening to the right, starting from \( x = 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
A parabola is a U-shaped curve that you often encounter when dealing with quadratic equations or inequalities. In this context, we are looking at a sideways-opening parabola described by the equation \( y^2 = x \). Understanding its shape and direction is critical.
Unlike the more common \( y = x^2 \) parabola, which opens upwards or downwards, this one opens sideways to the right because the \( x \) term depends on \( y \).
The vertex of this parabola is at the origin \((0, 0)\). From this point, it extends infinitely to the right. The parabola is symmetric about the x-axis.
Unlike the more common \( y = x^2 \) parabola, which opens upwards or downwards, this one opens sideways to the right because the \( x \) term depends on \( y \).
The vertex of this parabola is at the origin \((0, 0)\). From this point, it extends infinitely to the right. The parabola is symmetric about the x-axis.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we plot points using a pair of numerical coordinates. Each point on this plane is defined by a pair \((x, y)\). Here, \( x \) is the horizontal coordinate, and \( y \) is the vertical coordinate.
In analyzing quadratic inequalities like \( y^2 \leq x \), the coordinate plane allows us to visualize where certain inequalities hold true. It helps us segment the plane into different regions, such as areas "inside" or "on" a given curve, which in this case is a parabola.
In analyzing quadratic inequalities like \( y^2 \leq x \), the coordinate plane allows us to visualize where certain inequalities hold true. It helps us segment the plane into different regions, such as areas "inside" or "on" a given curve, which in this case is a parabola.
Solution Set
The solution set of an inequality like \( y^2 \leq x \) contains all the coordinate pairs \((x, y)\) that satisfy the inequality. In simpler terms, it includes every point where the inequality holds true.
For our inequality, the solution set lies *on and within* the parabola defined by \( y^2 = x \). This means every point \( (x, y) \) where the square of \( y \) is equal to or less than \( x \) is part of the solution set.
Visually, this set covers the region inside and along the parabola, extending from the parabola along the x-axis to the right.
For our inequality, the solution set lies *on and within* the parabola defined by \( y^2 = x \). This means every point \( (x, y) \) where the square of \( y \) is equal to or less than \( x \) is part of the solution set.
Visually, this set covers the region inside and along the parabola, extending from the parabola along the x-axis to the right.
Inequality Analysis
Analyzing inequalities involves understanding when one expression is less or more than another. For quadratic inequalities like \( y^2 \leq x \), the goal is to determine when \( y^2 \) does not exceed \( x \).
By rearranging the original inequality to \( y^2 \leq x \), we immediately recognize that the parabola \( y^2 = x \) serves as a boundary. Points along this curve satisfy the inequality, \( y^2 = x \), as do points below it, where \( y^2 < x \).
This analysis informs us about the regions on the coordinate plane where the inequality becomes valid, helping us delineate the solution set effectively.
By rearranging the original inequality to \( y^2 \leq x \), we immediately recognize that the parabola \( y^2 = x \) serves as a boundary. Points along this curve satisfy the inequality, \( y^2 = x \), as do points below it, where \( y^2 < x \).
This analysis informs us about the regions on the coordinate plane where the inequality becomes valid, helping us delineate the solution set effectively.
