What Is The Solution To $|2x - 8| \leq 6$?A. $1 \leq X \leq 7$ B. $x \leq -7$ Or $x \geq -1$ C. $x \leq 1$ Or $x \geq 7$ D. $-7 \leq X \leq -1$

Introduction

In mathematics, absolute value equations are a type of inequality that involves the absolute value of an expression. These equations are used to represent the distance of a value from zero on the number line. The absolute value of a number is its distance from zero, and it is always non-negative. In this article, we will explore the solution to the absolute value equation $|2x - 8| \leq 6$. We will break down the equation, analyze its components, and provide a step-by-step solution to find the values of x that satisfy the equation.

Understanding Absolute Value Equations

Absolute value equations are of the form $|ax + b| \leq c$ or $|ax + b| \geq c$, where a, b, and c are constants. The absolute value of an expression is its distance from zero on the number line. In the equation $|2x - 8| \leq 6$, the expression $2x - 8$ is inside the absolute value bars. To solve this equation, we need to consider two cases: when $2x - 8$ is positive and when $2x - 8$ is negative.

Case 1: $2x - 8 \geq 0$

When $2x - 8 \geq 0$, the absolute value equation becomes $2x - 8 \leq 6$. To solve this equation, we need to isolate the variable x. We can do this by adding 8 to both sides of the equation, which gives us $2x \leq 14$. Then, we can divide both sides of the equation by 2, which gives us $x \leq 7$.

Case 2: $2x - 8 < 0$

When $2x - 8 < 0$, the absolute value equation becomes $-(2x - 8) \leq 6$. To solve this equation, we need to simplify the left-hand side of the equation. We can do this by distributing the negative sign to the terms inside the parentheses, which gives us $-2x + 8 \leq 6$. Then, we can subtract 8 from both sides of the equation, which gives us $-2x \leq -2$. Finally, we can divide both sides of the equation by -2, which gives us $x \geq -1$.

Combining the Cases

Now that we have solved the two cases, we need to combine them to find the final solution. In Case 1, we found that $x \leq 7$, and in Case 2, we found that $x \geq -1$. Since these two cases are not mutually exclusive, we need to combine them to find the final solution. The final solution is the intersection of the two cases, which is $-1 \leq x \leq 7$.

Conclusion

In this article, we explored the solution to the absolute value equation $|2x - 8| \leq 6$. We broke down the equation, analyzed its components, and provided a step-by-step solution to find the values of x that satisfy the equation. We found that the final solution is $-1 \leq x \leq 7$. This solution represents the range of values of x that satisfy the equation.

Final Answer

The final answer is $-1 \leq x \leq 7$.

Introduction

In our previous article, we explored the solution to the absolute value equation $|2x - 8| \leq 6$. We broke down the equation, analyzed its components, and provided a step-by-step solution to find the values of x that satisfy the equation. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q&A Section

Q: What is the definition of an absolute value equation?

A: An absolute value equation is a type of inequality that involves the absolute value of an expression. The absolute value of a number is its distance from zero on the number line.

Q: How do I know when to use the positive or negative case in an absolute value equation?

A: To determine whether to use the positive or negative case, you need to consider the sign of the expression inside the absolute value bars. If the expression is positive, you use the positive case. If the expression is negative, you use the negative case.

Q: Can you explain the concept of the intersection of two cases?

A: Yes, the intersection of two cases is the set of values that satisfy both cases. In the case of the absolute value equation $|2x - 8| \leq 6$, the intersection of the two cases is $-1 \leq x \leq 7$.

Q: How do I know when to add or subtract a constant from both sides of an equation?

A: When adding or subtracting a constant from both sides of an equation, you need to make sure that you are not changing the direction of the inequality. If you are adding a positive constant, you need to add it to both sides of the equation. If you are subtracting a positive constant, you need to subtract it from both sides of the equation.

Q: Can you explain the concept of the distance of a value from zero on the number line?

A: Yes, the distance of a value from zero on the number line is the absolute value of the value. For example, the distance of 5 from zero on the number line is 5, and the distance of -5 from zero on the number line is also 5.

Q: How do I know when to use the distributive property to simplify an equation?

A: You should use the distributive property to simplify an equation when you have a negative sign in front of a term inside parentheses. For example, in the equation $-(2x - 8) \leq 6$, you would use the distributive property to simplify the left-hand side of the equation.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts or questions that readers may have about the solution to the absolute value equation $|2x - 8| \leq 6$. We hope that this Q&A section has been helpful in providing a better understanding of the concept of absolute value equations and how to solve them.

Final Answer

The final answer is $-1 \leq x \leq 7$.