Solve The Inequality: 3 K − 9 ≤ − 6 K − 225 3k - 9 \leq -6k - 225 3 K − 9 ≤ − 6 K − 225 Enter The Answer In The Space Provided. Use Numbers Instead Of Words.

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Introduction

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Linear inequalities are a fundamental concept in algebra, and solving them is a crucial skill for students to master. In this article, we will focus on solving the inequality $3k - 9 \leq -6k - 225$. We will break down the solution into manageable steps, using clear explanations and examples to help readers understand the process.

Understanding the Inequality

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Before we dive into solving the inequality, let's take a closer look at what it represents. The inequality $3k - 9 \leq -6k - 225$ is a linear inequality, which means it is an inequality that can be written in the form $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants, and $x$ is the variable.

In this case, the inequality is $3k - 9 \leq -6k - 225$. We can see that the left-hand side of the inequality is $3k - 9$, and the right-hand side is $-6k - 225$. Our goal is to solve for $k$.

Step 1: Add 9 to Both Sides

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To solve the inequality, we need to isolate the variable $k$ on one side of the inequality. We can start by adding 9 to both sides of the inequality. This will eliminate the negative term on the left-hand side.

3k - 9 + 9 ≤ -6k - 225 + 9

Simplifying the inequality, we get:

3k ≤ -6k - 216

Step 2: Add 6k to Both Sides

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Next, we need to get all the terms with $k$ on one side of the inequality. We can do this by adding 6k to both sides of the inequality.

3k + 6k ≤ -6k + 6k - 216

Simplifying the inequality, we get:

9k ≤ -216

Step 3: Divide Both Sides by 9

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Now that we have the inequality in the form $9k \leq -216$, we can solve for $k$ by dividing both sides of the inequality by 9.

9k / 9 ≤ -216 / 9

Simplifying the inequality, we get:

k ≤ -24

Conclusion

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In this article, we solved the linear inequality $3k - 9 \leq -6k - 225$ using a step-by-step approach. We added 9 to both sides of the inequality, then added 6k to both sides, and finally divided both sides by 9 to solve for $k$. The solution to the inequality is $k \leq -24$.

Tips and Tricks

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• When solving linear inequalities, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
• When adding or subtracting terms with variables, make sure to combine like terms to simplify the inequality.
• When dividing both sides of an inequality by a negative number, remember to flip the direction of the inequality.

Frequently Asked Questions

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• Q: What is the solution to the inequality $3k - 9 \leq -6k - 225$? A: The solution to the inequality is $k \leq -24$.
• Q: How do I solve a linear inequality? A: To solve a linear inequality, follow the steps outlined in this article: add or subtract terms to isolate the variable, then divide both sides by a non-zero constant to solve for the variable.
• 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 $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants, and $x$ is the variable. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.
  

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Introduction

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In our previous article, we solved the linear inequality $3k - 9 \leq -6k - 225$ using a step-by-step approach. In this article, we will provide a Q&A guide to help readers understand the concepts and techniques involved in solving linear inequalities.

Q&A: Solving Linear Inequalities

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Q: What is a linear inequality?

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

Q: How do I solve a linear inequality?

A: To solve a linear inequality, follow these steps:

1. Add or subtract terms to isolate the variable.
2. Divide both sides by a non-zero constant to solve for the variable.
3. Check your solution by plugging it back into the original inequality.

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 $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants, and $x$ is the variable. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I know which direction to flip the inequality when dividing both sides by a negative number?

A: When dividing both sides of an inequality by a negative number, you need to flip the direction of the inequality. For example, if you have the inequality $x \leq 5$ and you divide both sides by -2, the resulting inequality would be $x \geq -\frac{5}{2}$.

Q: Can I use the same steps to solve a quadratic inequality?

A: No, the steps for solving a quadratic inequality are different from those for solving a linear inequality. To solve a quadratic inequality, you need to factor the quadratic expression, if possible, and then use the sign of the quadratic expression to determine the solution.

Q: What is the solution to the inequality $2x + 5 \leq 3x - 2$?

A: To solve this inequality, we need to isolate the variable $x$. We can do this by subtracting 2x from both sides of the inequality, which gives us $5 \leq x - 2$. Then, we can add 2 to both sides of the inequality, which gives us $7 \leq x$. Therefore, the solution to the inequality is $x \geq 7$.

Q: What is the solution to the inequality $x^2 + 4x + 4 \leq 0$?

A: To solve this inequality, we need to factor the quadratic expression, if possible. In this case, we can factor the expression as $(x + 2)^2 \leq 0$. Since the square of any real number is always non-negative, the only way for the expression to be less than or equal to zero is if the expression is equal to zero. Therefore, the solution to the inequality is $x = -2$.

Tips and Tricks

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• When solving linear inequalities, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
• When adding or subtracting terms with variables, make sure to combine like terms to simplify the inequality.
• When dividing both sides of an inequality by a negative number, remember to flip the direction of the inequality.

Frequently Asked Questions

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• Q: What is the solution to the inequality $3k - 9 \leq -6k - 225$? A: The solution to the inequality is $k \leq -24$.
• Q: How do I solve a linear inequality? A: To solve a linear inequality, follow the steps outlined in this article: add or subtract terms to isolate the variable, then divide both sides by a non-zero constant to solve for the variable.
• 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 $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants, and $x$ is the variable. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Conclusion

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In this article, we provided a Q&A guide to help readers understand the concepts and techniques involved in solving linear inequalities. We covered topics such as the definition of a linear inequality, the steps for solving a linear inequality, and the difference between a linear inequality and a quadratic inequality. We also provided examples and tips to help readers practice and improve their skills.