If (\frac{3x-1}{2}\ge x+4) and (2x-7<5), what is the combined solution for (x)?

Correct answer: \(\varnothing\)

Multiplying the first inequality by 2 gives \(3x-1\ge 2x+8\), so \(x\ge 9\). From the second inequality, \(2x<12\), hence \(x<6\). No number can satisfy both \(x\ge 9\) and \(x<6\), so the combined solution is \(\varnothing\). The option \(x\ge 9\) solves only the first inequality, not both. Exam tip: for “and”, take the intersection of the solution sets.