What is the number line interval for (-2\le \frac{x-1}{3}<4)?

Correct answer: ([-5,13))

The direct answer is option A: \([-5,13)\). Start with \(-2\le\frac{x-1}{3}<4\). Since 3 is positive, multiply every part by 3 without changing either inequality sign: \(-6\le x-1<12\). Add 1 to all three parts: \(-5\le x<13\). The left inequality is non-strict, so \(-5\) is included and receives a square bracket. The right inequality is strict, so 13 is excluded and receives a round parenthesis. Therefore the interval is \([-5,13)\). Option A matches exactly. Option B, \((-5,13]\), reverses both endpoint inclusions. Option C, \([-7,11)\), results from incorrect arithmetic and does not follow from multiplying and adding. Option D, \((-7,11]\), has both wrong values and wrong endpoint types. A quick check: at \(x=-5\), the fraction is \(-2\), so the left boundary is included; at \(x=13\), it is 4, but the right inequality is strict, so 13 is excluded. Remember to multiply the entire chained inequality and preserve the signs when the multiplier is positive.