Solve The Inequality: $2|4-5x| \leq 9$A. $x \ \textless \ -\frac{1}{10}$ Or $x \ \textgreater \ \frac{17}{10}$B. $-\frac{1}{10} \ \textless \ X \ \textless \ \frac{17}{10}$C. $x \leq -\frac{1}{10}$

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve a specific type of inequality involving absolute values. The given inequality is $2|4-5x| \leq 9$, and our goal is to find the solution set for this inequality. We will break down the solution process step by step, using mathematical reasoning and logical deductions.

Understanding Absolute Value Inequalities

Before we dive into the solution, let's take a moment to understand what absolute value inequalities are. An absolute value inequality is an inequality that involves the absolute value of an expression. The absolute value of a number is its distance from zero on the number line, without considering whether it's positive or negative. In other words, the absolute value of a number is always non-negative.

Step 1: Isolate the Absolute Value Expression

To solve the inequality $2|4-5x| \leq 9$, we need to isolate the absolute value expression. We can do this by dividing both sides of the inequality by 2, which gives us $|4-5x| \leq \frac{9}{2}$.

Step 2: Write Two Separate Inequalities

Since the absolute value of an expression is always non-negative, we can write two separate inequalities to represent the original inequality:

$4-5x \leq \frac{9}{2}$ and $4-5x \geq -\frac{9}{2}$

Step 3: Solve the First Inequality

Let's solve the first inequality $4-5x \leq \frac{9}{2}$. We can start by subtracting 4 from both sides, which gives us $-5x \leq \frac{9}{2} - 4$. Simplifying the right-hand side, we get $-5x \leq \frac{1}{2}$.

Step 4: Solve for x in the First Inequality

To solve for x in the first inequality, we need to divide both sides by -5. However, since we are dividing by a negative number, we need to reverse the direction of the inequality sign. This gives us $x \geq -\frac{1}{10}$.

Step 5: Solve the Second Inequality

Now, let's solve the second inequality $4-5x \geq -\frac{9}{2}$. We can start by subtracting 4 from both sides, which gives us $-5x \geq -\frac{9}{2} - 4$. Simplifying the right-hand side, we get $-5x \geq -\frac{13}{2}$.

Step 6: Solve for x in the Second Inequality

To solve for x in the second inequality, we need to divide both sides by -5. Again, since we are dividing by a negative number, we need to reverse the direction of the inequality sign. This gives us $x \leq \frac{13}{10}$.

Step 7: Combine the Solutions

Now that we have solved both inequalities, we need to combine the solutions. The solution to the first inequality is $x \geq -\frac{1}{10}$, and the solution to the second inequality is $x \leq \frac{13}{10}$. However, we need to consider the original inequality $2|4-5x| \leq 9$, which is equivalent to $|4-5x| \leq \frac{9}{2}$. This means that the solution set must satisfy both inequalities simultaneously.

Step 8: Find the Intersection of the Solutions

To find the intersection of the solutions, we need to find the values of x that satisfy both inequalities simultaneously. This means that we need to find the values of x that are greater than or equal to $-\frac{1}{10}$ and less than or equal to $\frac{13}{10}$.

Step 9: Write the Final Solution

The final solution to the inequality $2|4-5x| \leq 9$ is $-\frac{1}{10} \leq x \leq \frac{13}{10}$. However, we need to consider the original inequality, which is equivalent to $|4-5x| \leq \frac{9}{2}$. This means that the solution set must satisfy the original inequality, which is equivalent to $2|4-5x| \leq 9$.

Conclusion

In this article, we have solved the inequality $2|4-5x| \leq 9$ step by step, using mathematical reasoning and logical deductions. We have broken down the solution process into several steps, including isolating the absolute value expression, writing two separate inequalities, solving each inequality, and combining the solutions. The final solution to the inequality is $-\frac{1}{10} \leq x \leq \frac{13}{10}$.

Final Answer

The final answer to the inequality $2|4-5x| \leq 9$ is:

A. $x \ \textless \ -\frac{1}{10}$ or $x \ \textgreater \ \frac{17}{10}$

This is incorrect, as the correct solution is $-\frac{1}{10} \leq x \leq \frac{13}{10}$.

B. $-\frac{1}{10} \ \textless \ x \ \textless \ \frac{17}{10}$

This is incorrect, as the correct solution is $-\frac{1}{10} \leq x \leq \frac{13}{10}$.

C. $x \leq -\frac{1}{10}$

This is incorrect, as the correct solution is $-\frac{1}{10} \leq x \leq \frac{13}{10}$.

The correct answer is:

D. $-\frac{1}{10} \leq x \leq \frac{13}{10}$

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Introduction

In our previous article, we solved the inequality $2|4-5x| \leq 9$ step by step, using mathematical reasoning and logical deductions. We broke down the solution process into several steps, including isolating the absolute value expression, writing two separate inequalities, solving each inequality, and combining the solutions. In this article, we will answer some frequently asked questions about the solution to the inequality.

Q: What is the final solution to the inequality $2|4-5x| \leq 9$?

A: The final solution to the inequality $2|4-5x| \leq 9$ is $-\frac{1}{10} \leq x \leq \frac{13}{10}$.

Q: Why do we need to isolate the absolute value expression?

A: We need to isolate the absolute value expression because it is the key to solving the inequality. By isolating the absolute value expression, we can write two separate inequalities that represent the original inequality.

Q: What are the two separate inequalities that we write?

A: The two separate inequalities that we write are $4-5x \leq \frac{9}{2}$ and $4-5x \geq -\frac{9}{2}$.

Q: How do we solve the first inequality $4-5x \leq \frac{9}{2}$?

A: To solve the first inequality, we start by subtracting 4 from both sides, which gives us $-5x \leq \frac{9}{2} - 4$. Simplifying the right-hand side, we get $-5x \leq \frac{1}{2}$. Then, we divide both sides by -5, which gives us $x \geq -\frac{1}{10}$.

Q: How do we solve the second inequality $4-5x \geq -\frac{9}{2}$?

A: To solve the second inequality, we start by subtracting 4 from both sides, which gives us $-5x \geq -\frac{9}{2} - 4$. Simplifying the right-hand side, we get $-5x \geq -\frac{13}{2}$. Then, we divide both sides by -5, which gives us $x \leq \frac{13}{10}$.

Q: How do we combine the solutions to the two inequalities?

A: To combine the solutions, we need to find the values of x that satisfy both inequalities simultaneously. This means that we need to find the values of x that are greater than or equal to $-\frac{1}{10}$ and less than or equal to $\frac{13}{10}$.

Q: What is the intersection of the solutions to the two inequalities?

A: The intersection of the solutions to the two inequalities is $-\frac{1}{10} \leq x \leq \frac{13}{10}$.

Q: Why is the solution to the inequality $2|4-5x| \leq 9$ important?

A: The solution to the inequality $2|4-5x| \leq 9$ is important because it provides a range of values for x that satisfy the inequality. This range of values can be used to solve a variety of problems in mathematics and other fields.

Q: Can you provide an example of how to use the solution to the inequality $2|4-5x| \leq 9$?

A: Yes, here is an example of how to use the solution to the inequality $2|4-5x| \leq 9$. Suppose we want to find the values of x that satisfy the inequality $2|4-5x| \leq 9$ and also satisfy the inequality $x \geq 2$. We can use the solution to the inequality $2|4-5x| \leq 9$ to find the values of x that satisfy both inequalities simultaneously.

Conclusion

In this article, we have answered some frequently asked questions about the solution to the inequality $2|4-5x| \leq 9$. We have provided step-by-step explanations of how to solve the inequality and have also provided examples of how to use the solution to the inequality. We hope that this article has been helpful in understanding the solution to the inequality $2|4-5x| \leq 9$.

Final Answer

The final answer to the inequality $2|4-5x| \leq 9$ is:

D. $-\frac{1}{10} \leq x \leq \frac{13}{10}$