\(x \ge -3\)


We follow the following roadmap:

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

• Find out whether the solutions are on the number line to the left or to the right of \(-3\).
  • First calculate the left- and right-hand sides of the inequality \(3x + 5 \ge x -1\) when you substitute a value of \(x\) less than or equal to \(-3\). For example, when you fill in \(x=-10\), then you get \(-25 \ge -11\) and this is a false statement. Any other value of \(x\) less than or equal to \(-3\) may be used too, and you still get a false statement.
  • Then calculate the left- and right-hand sides of the inequality \(3x + 5 \ge x -1\) when you substitute a value of \(x\) greater than or equal to \(-3\). For example, when you fill in \(x=10\), then you get \(35 \ge 9\) and this is a true statement. Any other value of \(x\) greater than or equal to \(-3\) may be used too, and you still get a true statement.
  • From these two numeric examples follows that solutions \(x\) of \(3x + 5 \ge x -1\) must satisfy \(x \ge -3\).
    The points where the inequality holds are shown in green in the number line below. An open circle around \(x=-3\) 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.