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Chapter 3: Problem 1

Is the ordered pair a solution to the given inequality? $$ y<5 x+1 ;(0,0) $$

Short Answer

Expert verified
Yes, the ordered pair (0, 0) is a solution.

Step by step solution

01

Identify the inequality and the ordered pair

We have the inequality \( y < 5x + 1 \) and the ordered pair \((0, 0)\). Our task is to determine if substituting \(x = 0\) and \(y = 0\) satisfies the inequality.
02

Substitute the values into the inequality

Replace \(x\) and \(y\) in the inequality with the coordinates from the ordered pair: \(0 < 5(0) + 1 \).
03

Simplify the expression

Calculate the expression on the right: \(5(0) + 1 = 0 + 1 = 1\).
04

Compare both sides

Compare \(0\) and \(1\) to see if the inequality \(0 < 1\) holds true.
05

Conclusion

Since \(0 < 1\) is true, the inequality is satisfied, meaning the ordered pair \((0, 0)\) is a solution to the inequality.

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Key Concepts

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

Understanding Ordered Pairs
Ordered pairs are fundamental in understanding points on a coordinate plane. Each ordered pair consists of two elements:
  • The first element is the x-coordinate.
  • The second element is the y-coordinate.
They are written in the form \((x, y)\).The order is crucial because it specifies the location on the plane:
  • The x-coordinate indicates the horizontal position, moving left or right.
  • The y-coordinate indicates the vertical position, moving up or down.
For example, the ordered pair \((0, 0)\) represents the origin, where both horizontal and vertical positions are zero.Understanding these pairs is key when determining if a set of points is a solution to an inequality or equation, as it describes specific points on the graph.
Finding an Inequality Solution
An inequality is a mathematical statement that shows the relationship of non-equality between two expressions. In this context, our task is to check if a specific point, given as an ordered pair, satisfies the inequality.When we talk about a solution to an inequality, we mean any point or set of points where the inequality holds true. For example, given the inequality \(y < 5x + 1\), a solution is any ordered pair \((x, y)\) that makes this statement true.In general, to find whether an ordered pair is a solution:
  • Identify the variables and inequality.
  • Substitute the values of the ordered pair into the inequality.
  • Perform any necessary calculations.
  • Check if the resulting statement is true.
If the inequality holds, then the ordered pair is part of the solution set.
How to Substitute Values
Substituting values is a straightforward method to test if a specific point meets the criteria set by an inequality or equation. Here's how you can do it:Assume you have the inequality \(y < 5x + 1\) and the ordered pair \((0, 0)\).Steps to substitute the values:
  • Replace the variable \(x\) with \(0\) from the ordered pair.
  • Similarly, replace \(y\) with \(0\).
This results in the equation:\(0 < 5(0) + 1\).Next, simplify the equation:
  • Calculate the value on the right side: \(5 \times 0 + 1 = 1\).
  • Compare the two sides of the inequality: \(0 < 1\).
Since \(0 < 1\) is true, the pair \((0, 0)\) satisfies the inequality. This process is essential for analyzing mathematical relationships with inequalities, ensuring clarity and accuracy in problem-solving.

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