Solve The Inequality For { A$} . . . { A + 6 \ \textless \ 12 \} Is { A = 5$}$ A Solution?

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Introduction

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In mathematics, inequalities are a fundamental concept that helps us compare values and make decisions. Solving inequalities involves finding the values of variables that satisfy a given inequality. In this article, we will focus on solving the inequality $a + 6 < 12$ and determine if $a = 5$ is a solution.

Understanding Inequalities

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Inequalities are mathematical statements that compare two values using greater than, less than, greater than or equal to, or less than or equal to. In this case, we have the inequality $a + 6 < 12$, which means that the value of $a$ plus 6 is less than 12.

Types of Inequalities

Linear Inequalities

Linear inequalities are inequalities that can be written in the form $ax + b < c$, where $a$, $b$, and $c$ are constants. In our example, the inequality $a + 6 < 12$ is a linear inequality.

Solving Linear Inequalities

To solve a linear inequality, we need to isolate the variable on one side of the inequality sign. In this case, we can subtract 6 from both sides of the inequality to get $a < 6$.

Properties of Inequalities

Inequalities have several properties that help us solve them. Some of the key properties include:

• Addition Property: If $a < b$, then $a + c < b + c$.
• Subtraction Property: If $a < b$, then $a - c < b - c$.
• Multiplication Property: If $a < b$ and $c > 0$, then $ac < bc$.
• Division Property: If $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$.

Solving the Inequality

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Now that we have a basic understanding of inequalities, let's solve the inequality $a + 6 < 12$. To do this, we can use the properties of inequalities to isolate the variable $a$.

Step 1: Subtract 6 from Both Sides

Subtracting 6 from both sides of the inequality gives us $a < 6$.

Step 2: Check the Solution

To check if $a = 5$ is a solution, we can substitute $a = 5$ into the inequality. If $a = 5$ satisfies the inequality, then it is a solution.

Checking the Solution

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Let's check if $a = 5$ is a solution to the inequality $a < 6$. We can substitute $a = 5$ into the inequality and see if it is true.

Substituting $a = 5$

Substituting $a = 5$ into the inequality gives us $5 < 6$. Since this is true, we can conclude that $a = 5$ is a solution to the inequality.

Conclusion

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In this article, we solved the inequality $a + 6 < 12$ and determined if $a = 5$ is a solution. We used the properties of inequalities to isolate the variable $a$ and found that $a < 6$. We then checked if $a = 5$ is a solution by substituting it into the inequality and found that it is indeed a solution.

Frequently Asked Questions

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Q: What is an inequality?

A: An inequality is a mathematical statement that compares two values using greater than, less than, greater than or equal to, or less than or equal to.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can use the properties of inequalities to do this.

Q: What are the properties of inequalities?

A: The properties of inequalities include the addition property, subtraction property, multiplication property, and division property.

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

A: To check if a solution is correct, you need to substitute the value into the inequality and see if it is true.

Final Thoughts

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Solving inequalities is an important concept in mathematics that helps us compare values and make decisions. By understanding the properties of inequalities and how to solve them, we can make informed decisions and solve real-world problems. In this article, we solved the inequality $a + 6 < 12$ and determined if $a = 5$ is a solution. We hope that this article has helped you understand how to solve inequalities and make informed decisions.

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Introduction

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In our previous article, we solved the inequality $a + 6 < 12$ and determined if $a = 5$ is a solution. In this article, we will continue to explore the concept of inequality solutions and answer some frequently asked questions.

Q&A: Inequality Solutions

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Q: What is an inequality solution?

A: An inequality solution is a value of the variable that satisfies the inequality.

Q: How do I find the solution to an inequality?

A: To find the solution to an inequality, you need to isolate the variable on one side of the inequality sign. You can use the properties of inequalities to do this.

Q: What are the steps to solve an inequality?

A: The steps to solve an inequality are:

1. Isolate the variable: Move all terms with the variable to one side of the inequality sign.
2. Simplify the inequality: Simplify the inequality by combining like terms.
3. Check the solution: Check if the solution satisfies the inequality.

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

A: To check if a solution is correct, you need to substitute the value into the inequality and see if it is true.

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

A: Some common mistakes to avoid when solving inequalities include:

• Not isolating the variable: Failing to isolate the variable on one side of the inequality sign.
• Not simplifying the inequality: Failing to simplify the inequality by combining like terms.
• Not checking the solution: Failing to check if the solution satisfies the inequality.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to:

1. Determine the direction of the inequality: Determine if the inequality is less than or greater than.
2. Plot the boundary: Plot the boundary of the inequality on the number line.
3. Shade the region: Shade the region that satisfies the inequality.

Q: What are some real-world applications of inequality solutions?

A: Some real-world applications of inequality solutions include:

• Finance: Inequality solutions are used to determine the minimum or maximum amount of money that can be borrowed or invested.
• Science: Inequality solutions are used to determine the minimum or maximum amount of a substance that can be present in a solution.
• Engineering: Inequality solutions are used to determine the minimum or maximum amount of a material that can be used in a design.

Conclusion

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In this article, we answered some frequently asked questions about inequality solutions. We hope that this article has helped you understand how to solve inequalities and make informed decisions. Remember to always isolate the variable, simplify the inequality, and check the solution to ensure that you are getting the correct answer.

Frequently Asked Questions

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Q: What is the difference between an inequality and an equation?

A: An inequality is a mathematical statement that compares two values using greater than, less than, greater than or equal to, or less than or equal to. An equation is a mathematical statement that states that two values are equal.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the values of the variables that satisfy all of the inequalities.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

• Linear inequalities: Inequalities that can be written in the form $ax + b < c$.
• Quadratic inequalities: Inequalities that can be written in the form $ax^2 + bx + c < 0$.
• Absolute value inequalities: Inequalities that involve absolute values.

Final Thoughts

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Inequality solutions are an important concept in mathematics that helps us compare values and make decisions. By understanding how to solve inequalities and answer frequently asked questions, we can make informed decisions and solve real-world problems. We hope that this article has helped you understand how to solve inequalities and make informed decisions.