Does The Point { (-2, 5)$}$ Make The Inequality ${ 2x + 7y \geq 31\$} True? You Must Show Your Work Or Explain Your Reasoning.

Introduction

In mathematics, inequalities are used to describe relationships between variables. A point is said to make an inequality true if it satisfies the condition described by the inequality. In this article, we will explore whether the point ${(-2, 5)}$ makes the inequality ${2x + 7y \geq 31}$ true.

Understanding the Inequality

The given inequality is ${2x + 7y \geq 31}$. This is a linear inequality in two variables, x and y. To determine whether the point ${(-2, 5)}$ makes this inequality true, we need to substitute the values of x and y into the inequality and check if it satisfies the condition.

Substituting the Values

To substitute the values of x and y into the inequality, we replace x with -2 and y with 5. The inequality becomes:

${2(-2) + 7(5) \geq 31}$

Simplifying the Expression

Now, we simplify the expression by evaluating the products and adding the results:

${2(-2) = -4}$ ${7(5) = 35}$

Substituting these values back into the inequality, we get:

${-4 + 35 \geq 31}$

Evaluating the Inequality

Now, we evaluate the inequality by adding -4 and 35:

${31 \geq 31}$

Conclusion

Since ${31 \geq 31}$, the inequality is true. Therefore, the point ${(-2, 5)}$ makes the inequality ${2x + 7y \geq 31}$ true.

Why is this Important?

Understanding whether a point makes an inequality true is crucial in various mathematical applications, such as graphing linear inequalities, solving systems of linear inequalities, and optimizing functions. In this article, we demonstrated how to substitute values into an inequality, simplify the expression, and evaluate the inequality to determine whether a point makes it true.

Real-World Applications

In real-world applications, inequalities are used to model various situations, such as:

• Finance: Inequalities are used to model financial transactions, such as investments and loans.
• Science: Inequalities are used to model scientific phenomena, such as population growth and chemical reactions.
• Engineering: Inequalities are used to model engineering problems, such as designing bridges and buildings.

Tips and Tricks

When working with inequalities, remember to:

• Substitute values carefully: Make sure to substitute the correct values into the inequality.
• Simplify the expression: Simplify the expression by evaluating products and adding results.
• Evaluate the inequality: Evaluate the inequality by comparing the result to the original inequality.

By following these tips and tricks, you can confidently determine whether a point makes an inequality true.

Conclusion

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

A: A linear equation is an equation in which the highest power of the variable(s) is 1. For example, ${2x + 3y = 5}$ is a linear equation. A linear inequality, on the other hand, is an inequality in which the highest power of the variable(s) is 1. For example, ${2x + 3y \geq 5}$ is a linear inequality.

Q: How do I determine whether a point makes a linear inequality true?

A: To determine whether a point makes a linear inequality true, you need to substitute the values of the variables into the inequality and check if it satisfies the condition. If the result is greater than or equal to the original inequality, then the point makes the inequality true.

Q: What is the significance of the greater than or equal to symbol (${\geq}$) in a linear inequality?

A: The greater than or equal to symbol (${\geq}$) in a linear inequality indicates that the result of the inequality can be equal to or greater than the original inequality. This means that the point can make the inequality true if the result is equal to or greater than the original inequality.

Q: Can a point make a linear inequality false?

A: Yes, a point can make a linear inequality false. If the result of the inequality is less than the original inequality, then the point makes the inequality false.

Q: How do I graph a linear inequality on a coordinate plane?

A: To graph a linear inequality on a coordinate plane, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality. If the inequality is of the form ${y \geq mx + b}$, then you need to graph the line ${y = mx + b}$ and shade the region above the line. If the inequality is of the form ${y \leq mx + b}$, then you need to graph the line ${y = mx + b}$ and shade the region below the line.

Q: Can a linear inequality have multiple solutions?

A: Yes, a linear inequality can have multiple solutions. If the inequality is of the form ${y \geq mx + b}$, then the solution set is the region above the line ${y = mx + b}$. If the inequality is of the form ${y \leq mx + b}$, then the solution set is the region below the line ${y = mx + b}$.

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you need to find the solution set that satisfies all the inequalities in the system. You can use various methods, such as graphing, substitution, or elimination, to solve the system.

Q: Can a system of linear inequalities have multiple solutions?

A: Yes, a system of linear inequalities can have multiple solutions. If the system has multiple inequalities, then the solution set may be a region that satisfies all the inequalities.

Conclusion

In conclusion, linear inequalities are an important concept in mathematics that can be used to model various real-world situations. By understanding how to determine whether a point makes a linear inequality true, graph a linear inequality on a coordinate plane, and solve a system of linear inequalities, you can confidently apply linear inequalities to solve problems in mathematics and other fields.