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Chapter 14: Problem 2

Determine whether \((0,0)\) is a solution of the inequality. $$y>x+1$$

Short Answer

Expert verified
The point \((0,0)\) is not a solution to the inequality \(y>x+1\).

Step by step solution

01

Substitute the Given Point

Place the coordinates of the given point \((0,0)\) into the inequality. Substitute \(0\) for \(x\) and \(0\) for \(y\) in the inequality. Thus, it becomes \(0 > 0+1\).
02

Simplify the Inequality

After substitution, the inequality becomes \(0 > 1\). This statement is saying zero is greater than one, which is incorrect.
03

Conclusion

Since \(0\) is not greater than \(1\), the point \((0,0)\) is not a solution to the inequality \(y>x+1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Introduction to the Coordinate System
Imagine the coordinate system as a large piece of graph paper. It helps us to locate points in a plane using pairs of numbers, known as coordinates. Each point is defined by an ordered pair
  • The first number represents the x-coordinate (horizontal placement)
  • The second number represents the y-coordinate (vertical placement)
Using this system, we can plot any location, such as the popular origin point the point (0,0), where the x-axis and y-axis meet. This foundational concept is crucial for graphing and understanding how equations and inequalities relate to the real world.
Understanding Algebra Concepts in Inequalities
Algebra uses symbols and letters to represent numbers. This abstraction makes solving equations and inequalities easier and more flexible. Building on algebra concepts, inequalities are like equations but instead of symbols like '=' to show equality, they use
  • '>' for greater than
  • '<' for less than
  • '鈮�' for greater than or equal to
  • '鈮�' for less than or equal to
Inequalities show the relationships between different values and have a range of solutions, unlike most equations that offer a specific answer. This understanding helps when determining if a point belongs to a region described by an inequality.
The Process of Solving an Inequality
Solving an inequality often begins with substituting known values into the inequality. For instance, in our inequality, test whether (0,0) fits by inserting 0 for both x and y, transforming it into an expression we can simplify. This process looks like this: 鈥� Original inequality: \[ y > x + 1 \] 鈥� Substitute x=0 and y=0: \[ 0 > 0 + 1 \] 鈥� Simplified expression: \[ 0 > 1 \] Here, simplifying helps us check if the statement is true or false, guiding us to understand whether the point satisfies the inequality.
Employing Mathematical Reasoning
Mathematical reasoning is all about thinking logically to solve problems. In this example, when we simplify \( 0 > 1 \) after substituting the point into the inequality, we apply reasoning to determine its validity. Since 0 is not greater than 1, we conclude that the point \((0,0)\) does not satisfy the inequality \( y > x + 1 \).To polish our error-checking skills, it鈥檚 important to use steps like:
  • Careful substitution of values
  • Clear simplification of expressions
  • Logical analysis of results
Practicing these processes sharpens our ability to solve a range of mathematical problems with confidence.

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