\(x \gt 1\)


We follow the following roadmap:

• Get started with the corresponding equation \[3x -2 = -5x {\,+\,}6\]
• Solve this equation:
  
  • Get the terms with \(x\) on the left-hand side of the equation (by adding \(5x\) on both sides):
    \(3x -2 +5x = -5x {\,+\,}6 +5x\), which simplifies to \(8x -2 = 6\).
  • Then move the terms without \(x\) to the right (by adding \(2\) both sides):
    \(8x -2 +2 = 6 +2\), which simplifies to \(8x = 8\).
    
    • Next, divide the left- and right-hand side by the coefficient of \(x\) (which is here \(8\)); this gives \(x = \;\frac{8}{8}\).
  • So, the solution of the equation is \(x = {1}\).

• Find out whether the solutions are on the number line to the left or to the right of \(1\).
  • First calculate the left- and right-hand sides of the inequality \(3x -2 \gt -5x {\,+\,}6\) when you substitute a value of \(x\) less than \(1\). For example, when you fill in \(x=-10\), then you get \(-32 \gt 56\) and this is a false statement. Any other value of \(x\) less than \(1\) may be used too, and you still get a false statement.
  • Then calculate the left- and right-hand sides of the inequality \(3x -2 \gt -5x {\,+\,}6\) when you substitute a value of \(x\) greater than \(1\). For example, when you fill in \(x=10\), then you get \(28 \gt -44\) and this is a true statement. Any other value of \(x\) greater than \(1\) may be used too, and you still get a true statement.
  • From these two numeric examples follows that solutions \(x\) of \(3x -2 \gt -5x {\,+\,}6\) must satisfy \(x \gt 1\).
    The points where the inequality holds are shown in green in the number line below. An open circle around \(x=1\) indicates that we are dealing with an inequality of the type \(\lt\) or \(\gt\), where in this case the point itself is not a solution. A closed circle indicates an inequality of the type \(\le\) or \(\ge\), and then the point marked on the number line is element of the solution set.