If the distance between place B and place C is x miles, what is the range of all possible sizes for x?
Understand the Problem
The question is asking for the range of possible lengths for the distance x between place B and place C, based on the distances and geometry of the positions of three places the student visited.
Answer
The range for distance $x$ is given by $|a - b| < x < a + b$.
Answer for screen readers
The distance $x$ between place B and place C lies in the range: $$ |a - b| < x < a + b $$
Steps to Solve
- Identify Given Distances
Let's denote the distances provided:
- Distance from place A to place B: $AB = a$
- Distance from place A to place C: $AC = b$
- The distance between place B and place C will be called $BC = x$.
- Apply the Triangle Inequality Theorem
To find the possible range for $x$, we can use the triangle inequality theorem, which states that for any triangle with sides $a$, $b$, and $x$, the following must hold: $$ x < a + b $$ $$ x > |a - b| $$
- Set Up the Inequalities
From the triangle inequality, we can set up our inequalities:
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For the upper bound: $$ x < a + b $$
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For the lower bound: $$ x > |a - b| $$
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Combine the Inequalities
We can summarize the range of possible distances $x$ by combining both inequalities: $$ |a - b| < x < a + b $$
- Substitute Actual Values (if available)
If actual values for $a$ and $b$ are provided in the problem, substitute them into the combined inequality to find the specific range for $x$.
The distance $x$ between place B and place C lies in the range: $$ |a - b| < x < a + b $$
More Information
The triangle inequality theorem is essential in geometry and is used to determine the possible lengths of a side in triangles given the other two sides. This can apply to various problems in navigation and geometry.
Tips
- Ignoring Absolute Value: A common mistake is not considering the absolute value for the lower bound, which can lead to incorrect results when $a < b$.
- Incorrectly Setting Boundaries: Failing to follow the triangle inequality strictly can give incorrect ranges.
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