Two of the most expensive cars in the world are car A and car B. The prices of these two cars differ by more than ​$15 comma 00015,000. The price of car A is ​$131 comma 645131,645.

Question

a. Assuming that you do not know which model is more​ expensive, write an absolute value inequality that describes this situation. Use x for the price of car B.
b. What are the possibilities for the price of car​ B?
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Part 1
a. Write an absolute value inequality that describes the given situation.

Answer 1

To describe the situation where the prices of car A and car B differ by more than $15,000, we can set up an absolute value inequality.

Let \( x \) be the price of car B, and we know the price of car A is $131,645. The condition states that the difference between the prices of car A and car B is more than $15,000. This can be formulated as:

\[ |x - 131645| > 15000 \]

This absolute value inequality states that the distance between the price of car B (x) and the price of car A ($131,645) is greater than $15,000.

Part a:

The absolute value inequality is:

\[ |x - 131645| > 15000 \]

Part b:

To solve for the possibilities for the price of car B (\( x \)), we can break the absolute value inequality into two separate inequalities:

1. \( x - 131645 > 15000 \)
2. \( x - 131645 < -15000 \) (which can be rewritten as \( x < 131645 - 15000 \))

Now let's solve both inequalities:

1. For \( x - 131645 > 15000 \): \[ x > 131645 + 15000 \ x > 146645 \]

2. For \( x - 131645 < -15000 \): \[ x < 131645 - 15000 \ x < 116645 \]

Conclusion:

The possibilities for the price of car B (\( x \)) are: \[ x < 116645 \quad \text{or} \quad x > 146645 \]

This means car B can either be priced less than $116,645 or more than $146,645.