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The Smiths and the Johnsons were competing in the final leg of the Amazing Race. In their race to the finish, the Smiths immediately took off on a 165 kilometer path traveling at an average speed of v kilometers per hour. The Johnsons' start was delayed by (1)/(2) hour. Eventually, they took a 180 kilometer path to the finish, traveling at an average speed that was 20 kilometers per hour faster than the Smiths' speed. The Johnsons arrived at the finish line first and won the race! Write an inequality in terms of v that models the situation.

The Smiths and the Johnsons were competing in the final leg of the Amazing Race. In their race to the finish, the Smiths immediately took off on a 165165 kilometer path traveling at an average speed of v v kilometers per hour.\newlineThe Johnsons' start was delayed by 12 \frac{1}{2} hour. Eventually, they took a 180180 kilometer path to the finish, traveling at an average speed that was 2020 kilometers per hour faster than the Smiths' speed.\newlineThe Johnsons arrived at the finish line first and won the race!\newlineWrite an inequality in terms of v v that models the situation.

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Q. The Smiths and the Johnsons were competing in the final leg of the Amazing Race. In their race to the finish, the Smiths immediately took off on a 165165 kilometer path traveling at an average speed of v v kilometers per hour.\newlineThe Johnsons' start was delayed by 12 \frac{1}{2} hour. Eventually, they took a 180180 kilometer path to the finish, traveling at an average speed that was 2020 kilometers per hour faster than the Smiths' speed.\newlineThe Johnsons arrived at the finish line first and won the race!\newlineWrite an inequality in terms of v v that models the situation.
  1. Denoting the average speed: Let's denote the average speed of the Smiths as vv kilometers per hour. The Smiths travel a distance of 165165 kilometers.\newlineThe time it takes for the Smiths to complete their path is given by the formula:\newlineTime == Distance // Speed\newlineSo for the Smiths, the time is:\newlineTimeSmiths=165v_{\text{Smiths}} = \frac{165}{v}
  2. Calculating the Smiths' time: The Johnsons' average speed is 2020 kilometers per hour faster than the Smiths', so their speed is v+20v + 20 kilometers per hour. They travel a distance of 180180 kilometers.\newlineThe time it takes for the Johnsons to complete their path is given by the same formula:\newlineTime == Distance // Speed\newlineSo for the Johnsons, the time is:\newlineTimeJohnsons=180v+20\text{Time}_{\text{Johnsons}} = \frac{180}{v + 20}
  3. Determining the Johnsons' speed: The Johnsons started 12\frac{1}{2} hour later than the Smiths. To account for this delay, we add 12\frac{1}{2} hour to the Johnsons' travel time.\newlineAdjusted TimeJohnsons=TimeJohnsons+12\text{Time}_{\text{Johnsons}} = \text{Time}_{\text{Johnsons}} + \frac{1}{2}\newlineAdjusted TimeJohnsons=180v+20+12\text{Time}_{\text{Johnsons}} = \frac{180}{v + 20} + \frac{1}{2}
  4. Calculating the Johnsons' time: The Johnsons arrived at the finish line first, which means their adjusted time must be less than the time it took the Smiths.\newlineSo the inequality that models the situation is:\newlineAdjusted TimeJohnsons_{Johnsons} < TimeSmiths_{Smiths}\newline\frac{180}{v + 20} + \frac{1}{2} < \frac{165}{v}

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