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A police officer uses a radar detector to determine that a motorist is traveling 34 mph in a 25-mph school zone. The driver goes to court and argues that the radar detector is not accurate. The manufacturer claims that the radar detector is calibrated to be in error by no more than 3 mph.

a. If \( x \) represents the motorist’s actual speed, write an inequality that represents an interval in which to estimate \( x \).

b. Solve the inequality and interpret the answer. Should the motorist receive a speeding ticket?

Answer :

a. This absolute value inequality is [tex]-3 \leq x - 34 \leq 3[/tex]

b. The motorist's actual speed is [tex][31, 37][/tex] mph.

Let's break down the problem step-by-step to solve the inequality and provide an interpretation.

Part (a): Writing the Inequality

Given:

  • The recorded speed by the radar detector: 34 mph
  • The margin of error of the radar detector: ±3 mph

If [tex]x[/tex] represents the motorist's actual speed, the inequality can be written as:

  • [tex]\|x - 34\| \leq 3[/tex]

This absolute value inequality can be split into two inequalities:

  • [tex]-3 \leq x - 34 \leq 3[/tex]

Part (b): Solving the Inequality

Using the above inequalities:

Adding 34 to each part of the inequality:

  • [tex]-3 + 34 \leq x \leq 3 + 34[/tex]

Simplifying:

  • [tex]31 \leq x \leq 37[/tex]

Therefore, the motorist's actual speed [tex]x[/tex] could be anywhere in the interval [tex][31, 37][/tex] mph.

Interpretation

  • The school zone speed limit is 25 mph. Even with the margin of error, the motorist's minimum possible speed is 31 mph, which is still above 25 mph.

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Rewritten by : Barada