Solve The Inequality:${-3(4x - 3) + 3 \leq 3}$
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Introduction
In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. Inequalities are mathematical statements that compare two expressions, indicating that one is greater than, less than, or equal to the other. 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 given inequality ${-3(4x - 3) + 3 \leq 3}$ as an example to demonstrate the step-by-step process of solving linear inequalities.
What are Linear Inequalities?
Linear inequalities are mathematical statements that compare two expressions, indicating that one is greater than, less than, or equal to the other. They can be written in the form of a linear equation, with the only difference being the use of the inequality symbol (>, <, or ≤) instead of the equality symbol (=). Linear inequalities can be classified into two types: strict inequalities and non-strict inequalities.
Strict Inequalities
Strict inequalities are inequalities that use the strict inequality symbols (>, <). They indicate that one expression is strictly greater than or strictly less than the other. For example, the inequality is a strict inequality.
Non-Strict Inequalities
Non-strict inequalities are inequalities that use the non-strict inequality symbols (≥, ≤). They indicate that one expression is greater than or equal to or less than or equal to the other. For example, the inequality is a non-strict inequality.
Solving Linear Inequalities
Solving linear inequalities involves isolating the variable on one side of the inequality symbol. The steps involved in solving linear inequalities are similar to those involved in solving linear equations, with the only difference being the use of the inequality symbol.
Step 1: Simplify the Inequality
The first step in solving a linear inequality is to simplify the inequality by combining like terms. This involves removing any parentheses and combining any like terms on the same side of the inequality symbol.
Step 2: Isolate the Variable
The next step is to isolate the variable on one side of the inequality symbol. This involves adding or subtracting the same value to both sides of the inequality to eliminate any constants on the same side as the variable.
Step 3: Solve for the Variable
Once the variable is isolated, the next step is to solve for the variable. This involves dividing both sides of the inequality by the coefficient of the variable to isolate the variable.
Solving the Given Inequality
Now that we have discussed the steps involved in solving linear inequalities, let's apply these steps to the given inequality ${-3(4x - 3) + 3 \leq 3}$.
Step 1: Simplify the Inequality
The first step is to simplify the inequality by combining like terms.
Step 2: Isolate the Variable
The next step is to isolate the variable on one side of the inequality symbol.
Step 3: Solve for the Variable
Once the variable is isolated, the next step is to solve for the variable.
Conclusion
In conclusion, solving linear inequalities involves isolating the variable on one side of the inequality symbol. The steps involved in solving linear inequalities are similar to those involved in solving linear equations, with the only difference being the use of the inequality symbol. By following the steps outlined in this article, you can solve linear inequalities with ease.
Frequently Asked Questions
Q: What is a linear inequality?
A: A linear inequality is a mathematical statement that compares two expressions, indicating that one is greater than, less than, or equal to the other.
Q: What are the types of linear inequalities?
A: There are two types of linear inequalities: strict inequalities and non-strict inequalities.
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 symbol by simplifying the inequality, isolating the variable, and solving for the variable.
Q: What is the difference between a linear equation and a linear inequality?
A: The main difference between a linear equation and a linear inequality is the use of the inequality symbol (>, <, or ≤) instead of the equality symbol (=).
References
- [1] "Linear Inequalities" by Math Open Reference
- [2] "Solving Linear Inequalities" by Khan Academy
- [3] "Linear Inequalities" by Wolfram MathWorld
Further Reading
If you want to learn more about linear inequalities, I recommend checking out the following resources:
- "Linear Inequalities" by Math Open Reference
- "Solving Linear Inequalities" by Khan Academy
- "Linear Inequalities" by Wolfram MathWorld
I hope this article has helped you understand and solve linear inequalities. If you have any questions or need further clarification, please don't hesitate to ask.
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Introduction
In our previous article, we discussed the concept of linear inequalities and provided a step-by-step guide on how to solve them. However, we understand that there may be some questions and doubts that you may have. In this article, we will address some of the frequently asked questions about linear inequalities and provide answers to help you better understand the concept.
Q&A
Q: What is a linear inequality?
A: A linear inequality is a mathematical statement that compares two expressions, indicating that one is greater than, less than, or equal to the other.
Q: What are the types of linear inequalities?
A: There are two types of linear inequalities: strict inequalities and non-strict inequalities.
Q: What is the difference between a linear equation and a linear inequality?
A: The main difference between a linear equation and a linear inequality is the use of the inequality symbol (>, <, or ≤) instead of the equality symbol (=).
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 symbol by simplifying the inequality, isolating the variable, and solving for the variable.
Q: What is the order of operations when solving a linear inequality?
A: The order of operations when solving a linear inequality is the same as when solving a linear equation: parentheses, exponents, multiplication and division, and addition and subtraction.
Q: Can I use the same methods to solve a linear inequality as I would to solve a linear equation?
A: Yes, you can use the same methods to solve a linear inequality as you would to solve a linear equation, with the exception of the inequality symbol.
Q: How do I know if an inequality is strict or non-strict?
A: If the inequality symbol is (>, <), it is a strict inequality. If the inequality symbol is (≥, ≤), it is a non-strict inequality.
Q: Can I add or subtract the same value to both sides of an inequality?
A: Yes, you can add or subtract the same value to both sides of an inequality, but you must also add or subtract the same value to the other side of the inequality.
Q: Can I multiply or divide both sides of an inequality by the same value?
A: Yes, you can multiply or divide both sides of an inequality by the same value, but you must also multiply or divide the other side of the inequality by the same 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 a linear equation, while a quadratic inequality is an inequality that can be written in the form of a quadratic equation.
Q: How do I solve a quadratic inequality?
A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the factored form to solve the inequality.
Additional Tips and Tricks
Tip 1: Always read the inequality carefully and make sure you understand what it is saying.
Tip 2: Use the order of operations to simplify the inequality before solving it.
Tip 3: Isolate the variable on one side of the inequality symbol by adding or subtracting the same value to both sides.
Tip 4: Solve for the variable by multiplying or dividing both sides of the inequality by the same value.
Tip 5: Check your answer by plugging it back into the original inequality.
Conclusion
In conclusion, linear inequalities are an important concept in mathematics that can be used to solve a wide range of problems. By understanding the concept of linear inequalities and following the steps outlined in this article, you can solve linear inequalities with ease. Remember to always read the inequality carefully, use the order of operations, isolate the variable, and solve for the variable.
Frequently Asked Questions (FAQs)
Q: What is a linear inequality?
A: A linear inequality is a mathematical statement that compares two expressions, indicating that one is greater than, less than, or equal to the other.
Q: What are the types of linear inequalities?
A: There are two types of linear inequalities: strict inequalities and non-strict inequalities.
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 symbol by simplifying the inequality, isolating the variable, and solving for the variable.
References
- [1] "Linear Inequalities" by Math Open Reference
- [2] "Solving Linear Inequalities" by Khan Academy
- [3] "Linear Inequalities" by Wolfram MathWorld
Further Reading
If you want to learn more about linear inequalities, I recommend checking out the following resources:
- "Linear Inequalities" by Math Open Reference
- "Solving Linear Inequalities" by Khan Academy
- "Linear Inequalities" by Wolfram MathWorld
I hope this article has helped you understand and solve linear inequalities. If you have any questions or need further clarification, please don't hesitate to ask.