Which is the correct number-line representation of ( 1\le 3x-2<16 )?

Correct answer: ( [1,6) )

The direct answer is option A: \([1,6)\). Solve the compound inequality as one chain: \(1\le3x-2<16\). Add 2 to all three parts: \(3\le3x<18\). Divide all parts by positive 3, so the signs do not change: \(1\le x<6\). Since 1 is included, use a closed endpoint at 1. Since 6 is excluded by the strict sign \(<\), use an open endpoint at 6. Option A, \([1,6)\), matches exactly. Option B reverses which endpoint is included. Option C is the interval before dividing by 3 and therefore gives values for \(3x\), not for \(x\). Option D has both the wrong interval and wrong endpoint inclusion. A useful check is to test \(x=2\): it satisfies the original inequality, while \(x=6\) gives \(16<16\), false. Memory cue: adding or dividing by a positive number keeps signs; equality makes a closed endpoint.