Solve $|2x - 4| \leq 8$.A. $x \geq -2$ And $x \leq 6$B. $x \geq -2$ And $x \leq 4$C. $x \geq -3$ Or $x \leq 2$D. $x \geq -2$ And $x \leq 5$

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Introduction

In this article, we will be solving the absolute value inequality $|2x - 4| \leq 8$. Absolute value inequalities are a type of mathematical problem that involves finding the values of a variable that satisfy an inequality involving absolute values. In this case, we are given the inequality $|2x - 4| \leq 8$, and we need to find the values of $x$ that satisfy this inequality.

Understanding Absolute Value Inequalities

Before we can solve the inequality $|2x - 4| \leq 8$, we need to understand what absolute value inequalities are and how to solve them. An absolute value inequality is an inequality that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of $-3$ is $3$, because $-3$ is $3$ units away from zero on the number line.

Solving Absolute Value Inequalities

To solve an absolute value inequality, we need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative. We can then use the properties of absolute values to simplify the inequality and solve for the variable.

Solving the Inequality $|2x - 4| \leq 8$

Now that we have a good understanding of absolute value inequalities, we can solve the inequality $|2x - 4| \leq 8$. To do this, we need to consider two cases: one where $2x - 4 \geq 0$, and one where $2x - 4 < 0$.

Case 1: $2x - 4 \geq 0$

In this case, we can rewrite the inequality as $2x - 4 \leq 8$. We can then add $4$ to both sides of the inequality to get $2x \leq 12$. Dividing both sides of the inequality by $2$, we get $x \leq 6$.

Case 2: $2x - 4 < 0$

In this case, we can rewrite the inequality as $-(2x - 4) \leq 8$. We can then simplify the inequality to get $-2x + 4 \leq 8$. Subtracting $4$ from both sides of the inequality, we get $-2x \leq 4$. Dividing both sides of the inequality by $-2$, we get $x \geq -2$.

Combining the Cases

Now that we have solved the inequality for both cases, we can combine the results to get the final solution. We have $x \leq 6$ in the first case, and $x \geq -2$ in the second case. Therefore, the final solution is $x \geq -2$ and $x \leq 6$.

Conclusion

In this article, we solved the absolute value inequality $|2x - 4| \leq 8$. We first understood the concept of absolute value inequalities and how to solve them. We then solved the inequality by considering two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative. We combined the results of the two cases to get the final solution, which is $x \geq -2$ and $x \leq 6$.

Final Answer

The final answer is A. $x \geq -2$ and $x \leq 6$.

Discussion

The solution to the inequality $|2x - 4| \leq 8$ is $x \geq -2$ and $x \leq 6$. This means that the values of $x$ that satisfy the inequality are all the numbers between $-2$ and $6$, inclusive.

Example

Let's consider an example to illustrate the solution. Suppose we want to find the values of $x$ that satisfy the inequality $|2x - 4| \leq 8$. We can use the solution we found earlier, which is $x \geq -2$ and $x \leq 6$. This means that the values of $x$ that satisfy the inequality are all the numbers between $-2$ and $6$, inclusive.

Graphical Representation

We can also represent the solution graphically. The solution $x \geq -2$ and $x \leq 6$ can be represented as a closed interval on the number line, with the endpoints $-2$ and $6$ included.

Real-World Applications

The solution to the inequality $|2x - 4| \leq 8$ has many real-world applications. For example, suppose we are given a budget of $8$ dollars to spend on a product that costs $2x - 4$ dollars. We want to find the values of $x$ that satisfy the inequality, which means we want to find the values of $x$ that allow us to spend at most $8$ dollars on the product.

Conclusion

In this article, we solved the absolute value inequality $|2x - 4| \leq 8$. We first understood the concept of absolute value inequalities and how to solve them. We then solved the inequality by considering two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative. We combined the results of the two cases to get the final solution, which is $x \geq -2$ and $x \leq 6$. We also discussed the real-world applications of the solution and represented the solution graphically.

Final Answer

The final answer is A. $x \geq -2$ and $x \leq 6$.

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Introduction

In our previous article, we solved the absolute value inequality $|2x - 4| \leq 8$. We found that the solution is $x \geq -2$ and $x \leq 6$. In this article, we will answer some common questions related to the solution of the inequality.

Q1: What is the meaning of the absolute value inequality $|2x - 4| \leq 8$?

A1: The absolute value inequality $|2x - 4| \leq 8$ means that the distance between $2x - 4$ and $0$ is less than or equal to $8$. In other words, $2x - 4$ is between $-8$ and $8$.

Q2: How do I solve an absolute value inequality?

A2: To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative. You can then use the properties of absolute values to simplify the inequality and solve for the variable.

Q3: What is the difference between $x \geq -2$ and $x \leq 6$ and $x \geq -2$ and $x \leq 4$?

A3: The difference between $x \geq -2$ and $x \leq 6$ and $x \geq -2$ and $x \leq 4$ is that the latter is a more restrictive solution. The solution $x \geq -2$ and $x \leq 4$ means that $x$ is between $-2$ and $4$, inclusive, while the solution $x \geq -2$ and $x \leq 6$ means that $x$ is between $-2$ and $6$, inclusive.

Q4: Can I use a calculator to solve an absolute value inequality?

A4: Yes, you can use a calculator to solve an absolute value inequality. However, you need to be careful when using a calculator to solve an absolute value inequality, as the calculator may not always give you the correct solution.

Q5: How do I graph the solution of an absolute value inequality?

A5: To graph the solution of an absolute value inequality, you need to plot the points that satisfy the inequality on a number line. You can then draw a closed interval on the number line to represent the solution.

Q6: What are some real-world applications of absolute value inequalities?

A6: Absolute value inequalities have many real-world applications. For example, they can be used to model the distance between two points, the amount of money spent on a product, or the number of people in a population.

Q7: Can I use absolute value inequalities to solve systems of equations?

A7: Yes, you can use absolute value inequalities to solve systems of equations. However, you need to be careful when using absolute value inequalities to solve systems of equations, as the solution may not always be unique.

Q8: How do I simplify an absolute value inequality?

A8: To simplify an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative. You can then use the properties of absolute values to simplify the inequality and solve for the variable.

Q9: What is the difference between an absolute value inequality and a linear inequality?

A9: The difference between an absolute value inequality and a linear inequality is that an absolute value inequality involves the absolute value of a variable or expression, while a linear inequality does not.

Q10: Can I use absolute value inequalities to solve quadratic equations?

A10: Yes, you can use absolute value inequalities to solve quadratic equations. However, you need to be careful when using absolute value inequalities to solve quadratic equations, as the solution may not always be unique.

Conclusion

In this article, we answered some common questions related to the solution of the absolute value inequality $|2x - 4| \leq 8$. We hope that this article has been helpful in clarifying any doubts you may have had about the solution of the inequality. If you have any further questions, please don't hesitate to ask.

Final Answer

The final answer is A. $x \geq -2$ and $x \leq 6$.