\(x \le 1\)


We follow the following roadmap:

• Get started with the corresponding equation \[x -1 = -2x {\,+\,}2\]
• Solve this equation:
  
  • Get the terms with \(x\) on the left-hand side of the equation (by adding \(2x\) on both sides):
    \(x -1 +2x = -2x {\,+\,}2 +2x\), which simplifies to \(3x -1 = 2\).
  • Then move the terms without \(x\) to the right (by adding \(1\) both sides):
    \(3x -1 +1 = 2 +1\), which simplifies to \(3x = 3\).
    
    • Next, divide the left- and right-hand side by the coefficient of \(x\) (which is here \(3\)); this gives \(x = \;\frac{3}{3}\).
  • 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 \(x -1 \le -2x {\,+\,}2\) when you substitute a value of \(x\) less than or equal to \(1\). For example, when you fill in \(x=-10\), then you get \(-11 \le 22\) and this is a true statement. Any other value of \(x\) less than or equal to \(1\) may be used too, and you still get a true statement.
  • Then calculate the left- and right-hand sides of the inequality \(x -1 \le -2x {\,+\,}2\) when you substitute a value of \(x\) greater than or equal to \(1\). For example, when you fill in \(x=10\), then you get \(9 \le -18\) and this is a false statement. Any other value of \(x\) greater than or equal to \(1\) may be used too, and you still get a false statement.
  • From these two numeric examples follows that solutions \(x\) of \(x -1 \le -2x {\,+\,}2\) must satisfy \(x \le 1\).
    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.