Add or subtract the same value to both sides of the inequality

Oct 13, 2025 by miodeya.com 148 views

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems in algebra, geometry, and other branches of mathematics. Inequalities are used to compare two or more quantities, and they can be either linear or non-linear. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of an equation with a variable and a constant. We will use the given inequality $x - 15 \leq -19$ as an example to demonstrate the steps involved in solving linear inequalities.

Understanding Linear Inequalities

Linear inequalities are inequalities that can be written in the form of an equation with a variable and a constant. They can be either greater than or equal to (≥) or less than or equal to (≤). For example, $x - 5 \geq 3$ and $2x + 1 \leq 7$ are both linear inequalities. Linear inequalities can be solved using various methods, including algebraic manipulation, graphical representation, and numerical methods.

Solving the Given Inequality

To solve the given inequality $x - 15 \leq -19$ , we need to isolate the variable $x$ on one side of the inequality. We can do this by adding 15 to both sides of the inequality. This will give us:

$x - 15 + 15 \leq -19 + 15$

Simplifying the inequality, we get:

$x \leq -4$

Explanation of the Solution

In the above solution, we added 15 to both sides of the inequality to isolate the variable $x$ . This is a valid operation because we are adding the same value to both sides of the inequality. By doing so, we are effectively canceling out the constant term on the left-hand side of the inequality, leaving us with the variable $x$ on one side.

Checking the Solution

To verify that our solution is correct, we can substitute $x = -4$ back into the original inequality. If the inequality holds true, then our solution is correct. Substituting $x = -4$ into the original inequality, we get:

$-4 - 15 \leq -19$

Simplifying the inequality, we get:

$-19 \leq -19$

This is a true statement, which confirms that our solution is correct.

Conclusion

In this article, we solved the inequality $x - 15 \leq -19$ using algebraic manipulation. We added 15 to both sides of the inequality to isolate the variable $x$ on one side. By doing so, we obtained the solution $x \leq -4$ . We also verified that our solution is correct by substituting $x = -4$ back into the original inequality.

Tips and Tricks for Solving Linear Inequalities

Here are some tips and tricks for solving linear inequalities:

Add or subtract the same value to both sides of the inequality: This is a valid operation that can be used to isolate the variable on one side of the inequality.
Multiply or divide both sides of the inequality by the same non-zero value: This is also a valid operation that can be used to isolate the variable on one side of the inequality.
Use inverse operations to isolate the variable: For example, if the inequality is in the form of $x + 5 \geq 3$ , we can subtract 5 from both sides of the inequality to isolate the variable $x$ .
Graphical representation: Linear inequalities can be represented graphically on a number line or a coordinate plane. This can be a useful tool for visualizing the solution set of the inequality.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving linear inequalities:

Not adding or subtracting the same value to both sides of the inequality: This can lead to an incorrect solution.
Not multiplying or dividing both sides of the inequality by the same non-zero value: This can also lead to an incorrect solution.
Not using inverse operations to isolate the variable: This can also lead to an incorrect solution.
Not checking the solution: This is an important step that can help verify that the solution is correct.

Real-World Applications of Linear Inequalities

Linear inequalities have numerous real-world applications in various fields, including:

Finance: Linear inequalities can be used to model financial problems, such as determining the minimum or maximum amount of money that can be invested or borrowed.
Science: Linear inequalities can be used to model scientific problems, such as determining the minimum or maximum temperature or pressure that can be achieved in a given experiment.
Engineering: Linear inequalities can be used to model engineering problems, such as determining the minimum or maximum stress or strain that can be applied to a given material.

Conclusion

In conclusion, solving linear inequalities is an important concept in mathematics that has numerous real-world applications. By understanding the steps involved in solving linear inequalities, we can apply this knowledge to solve a wide range of problems in various fields. We hope that this article has provided a clear and concise guide to solving linear inequalities, and we encourage readers to practice solving linear inequalities to become proficient in this skill.

Introduction

In our previous article, we discussed the concept of solving linear inequalities and provided a step-by-step guide to solving the inequality $x - 15 \leq -19$ . In this article, we will provide a Q&A guide to help readers understand and solve linear inequalities.

Q&A Guide

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of an equation with a variable and a constant. It can be either greater than or equal to (≥) or less than or equal to (≤).

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 to 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 difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form of an equation with a variable and a constant, whereas a quadratic inequality is an inequality that can be written in the form of a quadratic equation with a variable and a constant.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to plot the corresponding linear equation on a number line or a coordinate plane. Then, you need to shade the region that satisfies the inequality.

Q: What is the solution set of a linear inequality?

A: The solution set of a linear inequality is the set of all values of the variable that satisfy the inequality.

Q: How do I check if my solution is correct?

A: To check if your solution is correct, you need to substitute the solution back into the original inequality. If the inequality holds true, then your solution is correct.

Q: What are some common mistakes to avoid when solving linear inequalities?

A: Some common mistakes to avoid when solving linear inequalities include not adding or subtracting the same value to both sides of the inequality, not multiplying or dividing both sides of the inequality by the same non-zero value, not using inverse operations to isolate the variable, and not checking the solution.

Q: What are some real-world applications of linear inequalities?

A: Linear inequalities have numerous real-world applications in various fields, including finance, science, and engineering.

Q: How do I determine the minimum or maximum value of a linear inequality?

A: To determine the minimum or maximum value of a linear inequality, you need to isolate the variable on one side of the inequality and then determine the minimum or maximum value of the expression.

Q: What is the difference between a linear inequality and a system of linear inequalities?

A: A linear inequality is an inequality that can be written in the form of an equation with a variable and a constant, whereas a system of linear inequalities is a set of two or more linear inequalities that need to be satisfied simultaneously.

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 of each inequality and then find the intersection of the solution sets.

Conclusion

In conclusion, solving linear inequalities is an important concept in mathematics that has numerous real-world applications. By understanding the steps involved in solving linear inequalities, we can apply this knowledge to solve a wide range of problems in various fields. We hope that this Q&A guide has provided a clear and concise guide to understanding and solving linear inequalities.

Tips and Tricks for Solving Linear Inequalities

Here are some tips and tricks for solving linear inequalities:

Practice, practice, practice: The more you practice solving linear inequalities, the more comfortable you will become with the steps involved.
Use inverse operations: Inverse operations can be used to isolate the variable on one side of the inequality.
Graph the inequality: Graphing the inequality can help you visualize the solution set.
Check the solution: Checking the solution is an important step that can help verify that the solution is correct.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving linear inequalities:

Not adding or subtracting the same value to both sides of the inequality: This can lead to an incorrect solution.
Not multiplying or dividing both sides of the inequality by the same non-zero value: This can also lead to an incorrect solution.
Not using inverse operations to isolate the variable: This can also lead to an incorrect solution.
Not checking the solution: This is an important step that can help verify that the solution is correct.

Real-World Applications of Linear Inequalities

Linear inequalities have numerous real-world applications in various fields, including:

Finance: Linear inequalities can be used to model financial problems, such as determining the minimum or maximum amount of money that can be invested or borrowed.
Science: Linear inequalities can be used to model scientific problems, such as determining the minimum or maximum temperature or pressure that can be achieved in a given experiment.
Engineering: Linear inequalities can be used to model engineering problems, such as determining the minimum or maximum stress or strain that can be applied to a given material.