Evaluating the Ordered Pairs

Oct 13, 2025 by vinooalluri.com 119 views

Understanding Inequalities

In mathematics, an inequality is a statement that two expressions are not equal. It is often represented by the use of symbols such as <, >, ≤, or ≥. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation. We will use the inequality $3x + y > -8$ as an example to demonstrate how to find solutions to linear inequalities.

The Importance of Inequalities in Real-Life Scenarios

Inequalities are used to model real-life scenarios where we need to compare quantities. For instance, if we have a budget of $100 and we want to know how much we can spend on a particular item, we can use an inequality to represent the situation. In this case, the inequality would be $x < 100$ , where $x$ represents the amount we can spend on the item.

Solving the Inequality $3x + y > -8$

To solve the inequality $3x + y > -8$ , we need to isolate the variable $x$ . We can do this by subtracting $y$ from both sides of the inequality, which gives us $3x > -8 - y$ . Next, we can divide both sides of the inequality by $3$ , which gives us $x > \frac{-8 - y}{3}$ .

Evaluating the Ordered Pairs

Now that we have the solution to the inequality, we can evaluate the ordered pairs given in the problem. We will substitute the values of $x$ and $y$ from each ordered pair into the solution to the inequality and determine which pair satisfies the inequality.

Ordered Pair A: $(-3, 0)$

To determine if the ordered pair $(-3, 0)$ is a solution to the inequality, we can substitute the values of $x$ and $y$ into the solution to the inequality. We get $x > \frac{-8 - 0}{3}$ , which simplifies to $x > \frac{-8}{3}$ . Since $-3$ is not greater than $\frac{-8}{3}$ , the ordered pair $(-3, 0)$ is not a solution to the inequality.

Ordered Pair B: $(-2, -1)$

To determine if the ordered pair $(-2, -1)$ is a solution to the inequality, we can substitute the values of $x$ and $y$ into the solution to the inequality. We get $x > \frac{-8 - (-1)}{3}$ , which simplifies to $x > \frac{-7}{3}$ . Since $-2$ is greater than $\frac{-7}{3}$ , the ordered pair $(-2, -1)$ is a solution to the inequality.

Ordered Pair C: $(0, -10)$

To determine if the ordered pair $(0, -10)$ is a solution to the inequality, we can substitute the values of $x$ and $y$ into the solution to the inequality. We get $x > \frac{-8 - (-10)}{3}$ , which simplifies to $x > \frac{2}{3}$ . Since $0$ is not greater than $\frac{2}{3}$ , the ordered pair $(0, -10)$ is not a solution to the inequality.

Ordered Pair D: $(2, -16)$

To determine if the ordered pair $(2, -16)$ is a solution to the inequality, we can substitute the values of $x$ and $y$ into the solution to the inequality. We get $x > \frac{-8 - (-16)}{3}$ , which simplifies to $x > \frac{8}{3}$ . Since $2$ is not greater than $\frac{8}{3}$ , the ordered pair $(2, -16)$ is not a solution to the inequality.

Conclusion

In conclusion, the only ordered pair that is a solution to the inequality $3x + y > -8$ is the ordered pair $(-2, -1)$ . This is because the values of $x$ and $y$ in this pair satisfy the inequality, while the values of $x$ and $y$ in the other pairs do not.

Final Answer

The final answer is B. $(-2, -1)$ .

Understanding Inequalities

Q&A: Solving Inequalities

Q: What is the difference between a linear equation and a linear inequality?

A: A linear equation is a statement that two expressions are equal, while a linear inequality is a statement that two expressions are not equal. For example, the equation $2x + 3 = 5$ is a linear equation, while the inequality $2x + 3 > 5$ is a linear inequality.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What is the solution to a linear inequality?

A: The solution to a linear inequality is the set of all values of the variable that satisfy the inequality. For example, the solution to the inequality $x > 2$ is the set of all values of $x$ that are greater than 2.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality. For example, to graph the inequality $x > 2$ , you would graph the line $x = 2$ and then shade the region to the right of the line.

Q: Can a linear inequality have multiple solutions?

A: Yes, a linear inequality can have multiple solutions. For example, the inequality $x > 2$ has multiple solutions, including $x = 3$ , $x = 4$ , and $x = 5$ .

Q: How do I determine if a point is a solution to a linear inequality?

A: To determine if a point is a solution to a linear inequality, you need to substitute the values of the variables into the inequality and check if the inequality is true. For example, to determine if the point $(3, 4)$ is a solution to the inequality $x > 2$ , you would substitute $x = 3$ and $y = 4$ into the inequality and check if it is true.

Q: Can a linear inequality have no solutions?

A: Yes, a linear inequality can have no solutions. For example, the inequality $x < 2$ has no solutions, since there is no value of $x$ that is less than 2.

Conclusion

In conclusion, solving linear inequalities is an important concept in mathematics that has many real-life applications. By understanding how to solve linear inequalities, you can model and solve a wide range of problems in fields such as economics, finance, and science.

Final Answer

The final answer is that solving linear inequalities is a crucial skill that can be applied to a wide range of problems in mathematics and other fields.