please generate a video explanation for me---Title: Introduction to Linear Equations
Mastery: 100%
Question Number: 1/2
Question Stem:
If $\frac{x+6}{3} = \frac{x+6}{13}$, the value of $x+6$ is between which of the following pairs of values?
Options:
A. -7 and -3
B. -2 and 2
C. 2 and 7
D. 8 and 13
视频信息
答案文本
视频字幕
Welcome to this linear equation problem. We have the equation x plus 6 over 3 equals x plus 6 over 13. We need to determine which range contains the value of x plus 6. Let's examine the four given options carefully.
Let's analyze the structure of this equation. Notice that both fractions have exactly the same numerator, x plus 6. However, they have different denominators: 3 on the left side and 13 on the right side. This observation is crucial for solving the problem.
Here's the key insight: when two fractions have the same numerator but different denominators, they can only be equal if the numerator equals zero. Since x plus 6 appears in both numerators, we set x plus 6 equal to zero. Solving this gives us x equals negative 6, which means x plus 6 equals zero.
Now let's check each option to see which range contains zero. Option A is negative 7 to negative 3 - zero is not in this range. Option B is negative 2 to positive 2 - zero is definitely between these values. Option C is 2 to 7 - zero is not in this range. Option D is 8 to 13 - zero is not in this range either. Therefore, the correct answer is option B.
To summarize: we solved the equation by recognizing that when two fractions have the same numerator but different denominators, they can only be equal if the numerator equals zero. This gave us x plus 6 equals zero, which means the value of x plus 6 is zero. Among all the given options, only option B, negative 2 to positive 2, contains the value zero. Therefore, the correct answer is B.