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Chapter 3: Problem 101

Is the ordered pair a solution to the given inequality? $$ y>0 ;(-3,-1) $$

Short Answer

Expert verified
No, (-3, -1) is not a solution because \( y = -1 \) is not greater than 0.

Step by step solution

01

Understand the Inequality

The inequality is given as \( y > 0 \). This inequality describes the condition that the value of \( y \) must be greater than 0 for the ordered pair to be considered a solution.
02

Analyze the Ordered Pair

The ordered pair given is \((-3, -1)\), where \(x = -3\) and \(y = -1\). We need to check the value of \(y\) in this ordered pair with respect to the inequality \( y > 0 \).
03

Substitute in the Inequality

Substitute \(y = -1\) from the ordered pair into the inequality \( y > 0 \) to see if it holds true. So we evaluate whether \(-1 > 0\).
04

Evaluate the Result

The condition \(-1 > 0\) is false, as \(-1\) is less than \(0\), not greater. Therefore, the ordered pair \((-3, -1)\) does not satisfy the inequality \( y > 0 \).

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Key Concepts

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

Understanding Ordered Pairs in Inequalities
In math, an ordered pair \( (x, y) \) is a pair of numbers used to represent a point on a coordinate plane. The first number, \( x \), is the horizontal position, and the second number, \( y \), is the vertical position. To determine if an ordered pair is a solution to an inequality, we substitute the values into the given inequality. If the inequality holds true with these values, the pair is a solution. For example, with the ordered pair \( (-3, -1) \), \( x = -3 \) and \( y = -1 \). It's essential to clearly understand each part of the pair to properly evaluate whether it satisfies an inequality.
Evaluating Inequality Solutions
Solving inequalities involves determining which values satisfy the given condition. An inequality like \( y > 0 \) tells us that \( y \) must be greater than zero to be a solution. Let's break it down:
  • Identify the inequality: Understand what the inequality is asking. Here, \( y > 0 \) means we're searching for values of \( y \) that are positive.
  • Substitute the values: Plug \( y = -1 \) from the ordered pair into \( y > 0 \).
  • Evaluate: Check if the statement \(-1 > 0\) is true or false. In this case, it's false because \(-1\) is less than \(0\), not greater.
If the evaluation results in a false statement, the ordered pair is not a solution. For this example, \((-3, -1)\) does not satisfy \( y > 0 \).
Coordinate Plane Analysis
The coordinate plane is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It's crucial for analyzing inequalities. Each point on this plane is represented by an ordered pair, \( (x, y) \).When working with inequalities:
  • The inequality \( y > 0 \) refers to all points above the x-axis since points on the x-axis have \ y = 0 \.
  • To check an ordered pair like \((-3, -1)\), plot the point on the plane. If the point lies above the x-axis, it satisfies \( y > 0 \).
  • Visualizing on the graph helps understand why certain regions satisfy the inequality.
Thus, coordinate plane analysis provides a graphical way to visualize and confirm solutions of inequalities, like determining that \((-3, -1)\) does not lie above the x-axis, validating it doesn't satisfy \( y > 0 \).

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