Select all the solutions of \( 1<3 x-2<13 \). A. \( 11 \) and \( x<5 \) C. \( x<1 \) and \( x>5 \) D. \( 5>x>1 \) E. \( 5

Ask by Bradley Burton. in the United States

Mar 12,2025

First, let's break down the compound inequality \( 1 < 3x - 2 < 13 \) into two parts. 1. For the left part: \[ 1 < 3x - 2 \] Adding 2 to both sides: \[ 3 < 3x \] Dividing by 3: \[ 1 < x \] (or \( x > 1 \)) 2. For the right part: \[ 3x - 2 < 13 \] Adding 2 to both sides: \[ 3x < 15 \] Dividing by 3: \[ x < 5 \] Combining both results, we find: \[ 1 < x < 5 \] Thus, the solution can be expressed as \( x > 1 \) and \( x < 5 \), or simply \( 5 > x > 1 \). Now, the correct options from the choices given are: - A. \( 1 < x < 5 \) - B. \( x > 1 \) and \( x < 5 \) - D. \( 5 > x > 1 \) So the solutions are A, B, and D. C and E are incorrect as they do not satisfy the derived inequalities.