Consider The Inequality $-5x - 9 \leq 6$.Solve The Inequality.

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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, with a specific example of the inequality $-5x - 9 \leq 6$. We will break down the solution process into manageable steps, making it easier for readers to understand and apply the concepts.

Understanding Linear Inequalities

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A linear inequality is an inequality that can be written in the form $ax + b \leq c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The goal of solving a linear inequality is to isolate the variable $x$ on one side of the inequality sign.

Types of Linear Inequalities

Equalities and Inequalities

There are two types of linear inequalities: equalities and inequalities. An equality is a statement that two expressions are equal, while an inequality is a statement that one expression is greater than or less than another expression.

Solving Linear Inequalities

Adding or Subtracting Constants

To solve a linear inequality, we can add or subtract constants to both sides of the inequality. This will not change the direction of the inequality sign.

Example: Solving the Inequality $-5x - 9 \leq 6$

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Let's solve the inequality $-5x - 9 \leq 6$ using the steps outlined above.

Step 1: Add 9 to Both Sides

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First, we add 9 to both sides of the inequality to get rid of the negative term.

-5x - 9 + 9 ≤ 6 + 9
-5x ≤ 15

Step 2: Divide Both Sides by -5

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Next, we divide both sides of the inequality by -5. Remember to reverse the direction of the inequality sign when dividing by a negative number.

(-5x)/(-5) ≥ 15/(-5)
x ≥ -3

Step 3: Write the Solution in Interval Notation

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The solution to the inequality $-5x - 9 \leq 6$ is $x \geq -3$. We can write this in interval notation as $[-3, \infty)$.

Conclusion

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Solving linear inequalities requires a step-by-step approach. By adding or subtracting constants and dividing both sides by a non-zero number, we can isolate the variable and find the solution. In this article, we solved the inequality $-5x - 9 \leq 6$ using these steps. The solution is $x \geq -3$, which can be written in interval notation as $[-3, \infty)$.

Frequently Asked Questions

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

A: A linear inequality is an inequality that can be written in the form $ax + b \leq c$, while a linear equation is an equation that can be written in the form $ax + b = c$.

Q: How do I know which direction to reverse the inequality sign when dividing by a negative number?

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

Q: Can I use the same steps to solve a linear inequality with a greater than or equal to sign?

A: Yes, the same steps can be used to solve a linear inequality with a greater than or equal to sign. The only difference is that you will have to reverse the direction of the inequality sign when dividing by a negative number.

Final Thoughts

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Solving linear inequalities is a crucial skill for students to master. By following the steps outlined in this article, you can solve linear inequalities with ease. Remember to add or subtract constants, divide both sides by a non-zero number, and reverse the direction of the inequality sign when dividing by a negative number. With practice, you will become proficient in solving linear inequalities and be able to apply this skill to a wide range of mathematical problems.

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Introduction

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Solving linear inequalities can be a challenging task, but with the right guidance, it can become a breeze. In this article, we will address some of the most frequently asked questions about solving linear inequalities. Whether you are a student struggling to understand the concept or a teacher looking for ways to explain it to your students, this article is for you.

Q&A: Solving Linear Inequalities

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

A: A linear inequality is an inequality that can be written in the form $ax + b \leq c$, while a linear equation is an equation that can be written in the form $ax + b = c$.

Q: How do I know which direction to reverse the inequality sign when dividing by a negative number?

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

Q: Can I use the same steps to solve a linear inequality with a greater than or equal to sign?

A: Yes, the same steps can be used to solve a linear inequality with a greater than or equal to sign. The only difference is that you will have to reverse the direction of the inequality sign when dividing by a negative number.

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

A: To solve the inequality $-3x + 2 \geq 5$, we can start by subtracting 2 from both sides:

-3x + 2 - 2 ≥ 5 - 2
-3x ≥ 3

Next, we can divide both sides by -3 and reverse the direction of the inequality sign:

(-3x)/(-3) ≤ 3/(-3)
x ≤ -1

The solution to the inequality $-3x + 2 \geq 5$ is $x ≤ -1$.

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

A: To write the solution to a linear inequality in interval notation, you need to determine the values of the variable that satisfy the inequality. For example, if the solution to the inequality $x \geq -3$ is $x \geq -3$, you can write it in interval notation as $[-3, \infty)$.

Q: Can I use the same steps to solve a linear inequality with a less than or equal to sign?

A: Yes, the same steps can be used to solve a linear inequality with a less than or equal to sign. The only difference is that you will have to reverse the direction of the inequality sign when dividing by a negative number.

Q: What is the solution to the inequality $2x - 4 \leq 3$?

A: To solve the inequality $2x - 4 \leq 3$, we can start by adding 4 to both sides:

2x - 4 + 4 ≤ 3 + 4
2x ≤ 7

Next, we can divide both sides by 2:

(2x)/2 ≤ 7/2
x ≤ 3.5

The solution to the inequality $2x - 4 \leq 3$ is $x ≤ 3.5$.

Conclusion

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Solving linear inequalities can be a challenging task, but with the right guidance, it can become a breeze. By following the steps outlined in this article, you can solve linear inequalities with ease. Remember to add or subtract constants, divide both sides by a non-zero number, and reverse the direction of the inequality sign when dividing by a negative number. With practice, you will become proficient in solving linear inequalities and be able to apply this skill to a wide range of mathematical problems.

Final Thoughts

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Solving linear inequalities is a crucial skill for students to master. By following the steps outlined in this article, you can solve linear inequalities with ease. Remember to add or subtract constants, divide both sides by a non-zero number, and reverse the direction of the inequality sign when dividing by a negative number. With practice, you will become proficient in solving linear inequalities and be able to apply this skill to a wide range of mathematical problems.

Additional Resources

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If you are struggling to understand the concept of solving linear inequalities, there are many additional resources available to help you. Some of these resources include:

• Online tutorials and videos
• Math textbooks and workbooks
• Online math communities and forums
• Math tutors and instructors

By taking advantage of these resources, you can get the help and support you need to master the concept of solving linear inequalities.

Final Q&A

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Q: What is the final answer to the inequality $-5x - 9 \leq 6$?

A: The final answer to the inequality $-5x - 9 \leq 6$ is $x \geq -3$.

Q: Can I use the same steps to solve a linear inequality with a greater than sign?

A: Yes, the same steps can be used to solve a linear inequality with a greater than sign. The only difference is that you will have to reverse the direction of the inequality sign when dividing by a negative number.

Q: What is the solution to the inequality $x + 2 \geq 5$?

A: To solve the inequality $x + 2 \geq 5$, we can start by subtracting 2 from both sides:

x + 2 - 2 ≥ 5 - 2
x ≥ 3

The solution to the inequality $x + 2 \geq 5$ is $x ≥ 3$.