A traffic congestion index is modeled by \( t^2 - 6t + 8 \leq 0 \). Find the largest value of \( t \) that satisfies this inequality.

["Solving the Traffic Congestion Inequality: Finding the Largest Value of ( t ) Satisfying ( t^2 - 6t + 8 \leq 0 )", "Traffic congestion modeling is a critical aspect of urban planning and transportation engineering. Understanding when traffic delays exceed acceptable limits helps city planners develop smarter infrastructure and traffic management strategies. One powerful tool in this analysis is the quadratic inequality ( t^2 - 6t + 8 \leq 0 ), which helps identify time intervals ( t ) during which congestion exceeds modeled thresholds.", "### Understanding the Inequality", "The inequality ( t^2 - 6t + 8 \leq 0 ) represents the condition where traffic congestion levels remain at or above a critical threshold. Solving this inequality allows us to determine the exact time intervals (values of ( t )) when congestion is severe enough to require intervention.", "Start by factoring the quadratic expression:", "[\nt^2 - 6t + 8 = (t - 2)(t - 4)\n]", "So the inequality becomes:", "[\n(t - 2)(t - 4) \leq 0\n]", "### Solving the Inequality", "A product of two factors is less than or equal to zero when one factor is non-positive and the other is non-negative. This happens between the roots (inclusive):", "[\n2 \leq t \leq 4\n]", "Thus, traffic congestion modeled by this expression is saturated (or severe) during the time interval from ( t = 2 ) to ( t = 4 ).", "### Finding the Largest Value of ( t )", "From the solution interval ( [2, 4] ), the largest value of ( t ) satisfying the inequality is:", "[\nt = 4\n]", "### Conclusion", "The inequality ( t^2 - 6t + 8 \leq 0 ) proves that traffic congestion reaches critical levels between 2 and 4 time units, with the peak at ( t = 4 ). This insight is invaluable for scheduling traffic interventions, ramp metering, or public transport alerts. Recognizing the boundaries of congestion helps authorities proactively manage urban mobility and reduce delays.", "Key Takeaways:", "- The inequality models critical traffic congestion thresholds.\n- Solving ( t^2 - 6t + 8 \leq 0 ) yields ( t \in [2, 4] ).\n- The largest valid time is ( t = 4 ).", "By using such mathematical modeling, cities can effectively monitor and manage traffic flow, making commutes smoother and more predictable.", "---", "For further insights on applying quadratic models in urban traffic analysis, explore advanced statistical tools and GIS-based congestion mapping."]