Solve For $t$.$t - 2 \leq 18$Write The Solution As An Inequality (for Example, \$t \ \textgreater \ 9$[/tex\]).$\square$

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Introduction

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Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear inequalities of the form $ax \leq b$, where $a$ and $b$ are constants, and $x$ is the variable. We will use the given problem, $t - 2 \leq 18$, as a case study to demonstrate the step-by-step process of solving linear inequalities.

Understanding the Problem

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The given problem is a linear inequality of the form $t - 2 \leq 18$. Our goal is to solve for $t$ and write the solution as an inequality. To do this, we need to isolate the variable $t$ on one side of the inequality.

Step 1: Add 2 to Both Sides

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To isolate $t$, we need to get rid of the constant term $-2$ that is being subtracted from $t$. We can do this by adding $2$ to both sides of the inequality. This will give us:

$t - 2 + 2 \leq 18 + 2$

Step 2: Simplify the Inequality

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When we add $2$ to both sides of the inequality, the $-2$ and $+2$ cancel each other out, leaving us with:

$t \leq 20$

Step 3: Write the Solution as an Inequality

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Now that we have isolated $t$ on one side of the inequality, we can write the solution as an inequality. In this case, the solution is:

$t \leq 20$

Conclusion

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Solving linear inequalities is a straightforward process that involves isolating the variable on one side of the inequality. By following the steps outlined in this article, we can solve linear inequalities of the form $ax \leq b$ and write the solution as an inequality. In the next section, we will explore some examples of linear inequalities and practice solving them.

Examples and Practice

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Example 1: $x + 3 \leq 12$

To solve this inequality, we need to isolate $x$ on one side of the inequality. We can do this by subtracting $3$ from both sides of the inequality.

$x + 3 - 3 \leq 12 - 3$

This simplifies to:

$x \leq 9$

Example 2: $y - 4 \leq 15$

To solve this inequality, we need to isolate $y$ on one side of the inequality. We can do this by adding $4$ to both sides of the inequality.

$y - 4 + 4 \leq 15 + 4$

This simplifies to:

$y \leq 19$

Tips and Tricks

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• When solving linear inequalities, always isolate the variable on one side of the inequality.
• Use addition and subtraction to get rid of constant terms that are being added to or subtracted from the variable.
• Simplify the inequality by combining like terms.
• Write the solution as an inequality in the form $x \leq b$ or $x \geq b$.

Common Mistakes to Avoid

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• Failing to isolate the variable on one side of the inequality.
• Not simplifying the inequality by combining like terms.
• Writing the solution as an equation instead of an inequality.

Conclusion

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Solving linear inequalities is a crucial skill for students to master. By following the steps outlined in this article, we can solve linear inequalities of the form $ax \leq b$ and write the solution as an inequality. Remember to isolate the variable on one side of the inequality, use addition and subtraction to get rid of constant terms, simplify the inequality by combining like terms, and write the solution as an inequality in the form $x \leq b$ or $x \geq b$. With practice and patience, you will become proficient in solving linear inequalities and be able to tackle more complex problems with confidence.

Final Thoughts

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Solving linear inequalities is not just about following a set of steps; it's about understanding the underlying concepts and being able to apply them to real-world problems. By mastering the art of solving linear inequalities, you will be able to tackle a wide range of mathematical problems and develop a deeper understanding of the subject. So, take the time to practice and review, and you will be solving linear inequalities like a pro in no time!

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Introduction

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In our previous article, we explored the basics of solving linear inequalities and provided a step-by-step guide on how to solve them. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will address some of the most frequently asked questions about solving linear inequalities and provide detailed explanations and examples to help you understand the concepts better.

Q&A

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Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax \leq b$, where $a$ and $b$ are constants, and $x$ is the variable.

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.

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

A: A linear equation is an equation that can be written in the form $ax = b$, where $a$ and $b$ are constants, and $x$ is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form $ax \leq b$ or $ax \geq b$.

Q: How do I know which direction to add or subtract when solving a linear inequality?

A: When solving a linear inequality, you need to add or subtract the same value to both sides of the inequality. If the inequality is of the form $ax \leq b$, you need to add the value to the left side of the inequality. If the inequality is of the form $ax \geq b$, you need to add the value to the right side of the inequality.

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 make the inequality true. For example, if the inequality is $x \leq 5$, the solution is all values of $x$ that are less than or equal to 5.

Q: How do I write the solution to a linear inequality?

A: To write the solution to a linear inequality, you need to use the correct notation. If the inequality is of the form $ax \leq b$, you need to write the solution as $x \leq b$. If the inequality is of the form $ax \geq b$, you need to write the solution as $x \geq b$.

Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?

A: No, you cannot use the same steps to solve a linear inequality as you would to solve a linear equation. When solving a linear inequality, you need to isolate the variable on one side of the inequality, whereas when solving a linear equation, you need to isolate the variable on both sides of the equation.

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

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

• Failing to isolate the variable on one side of the inequality
• Not simplifying the inequality by combining like terms
• Writing the solution as an equation instead of an inequality

Examples and Practice

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Example 1: $x + 2 \leq 10$

To solve this inequality, we need to isolate $x$ on one side of the inequality. We can do this by subtracting 2 from both sides of the inequality.

$x + 2 - 2 \leq 10 - 2$

This simplifies to:

$x \leq 8$

Example 2: $y - 3 \geq 12$

To solve this inequality, we need to isolate $y$ on one side of the inequality. We can do this by adding 3 to both sides of the inequality.

$y - 3 + 3 \geq 12 + 3$

This simplifies to:

$y \geq 15$

Conclusion

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Solving linear inequalities is a crucial skill for students to master. By understanding the concepts and following the steps outlined in this article, you will be able to solve linear inequalities with confidence. Remember to isolate the variable on one side of the inequality, use addition and subtraction to get rid of constant terms, simplify the inequality by combining like terms, and write the solution as an inequality in the form $x \leq b$ or $x \geq b$. With practice and patience, you will become proficient in solving linear inequalities and be able to tackle more complex problems with ease.

Final Thoughts

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Solving linear inequalities is not just about following a set of steps; it's about understanding the underlying concepts and being able to apply them to real-world problems. By mastering the art of solving linear inequalities, you will be able to tackle a wide range of mathematical problems and develop a deeper understanding of the subject. So, take the time to practice and review, and you will be solving linear inequalities like a pro in no time!