\(x \le 0\)


We follow the following roadmap:

• Get started with the corresponding equation \[-x -4 = -2x -4\]
• Solve this equation:
  
  • Get the terms with \(x\) on the left-hand side of the equation (by adding \(2x\) on both sides):
    \(-x -4 +2x = -2x -4 +2x\), which simplifies to \(x -4 = -4\).
  • Then move the terms without \(x\) to the right (by adding \(4\) both sides):
    \(x -4 +4 = -4 +4\), which simplifies to \(x = 0\).
    
  • So, the solution of the equation is \(x = {0}\).

• Find out whether the solutions are on the number line to the left or to the right of \(0\).
  • First calculate the left- and right-hand sides of the inequality \(-x -4 \le -2x -4\) when you substitute a value of \(x\) less than or equal to \(0\). For example, when you fill in \(x=-10\), then you get \(6 \le 16\) and this is a true statement. Any other value of \(x\) less than or equal to \(0\) may be used too, and you still get a true statement.
  • Then calculate the left- and right-hand sides of the inequality \(-x -4 \le -2x -4\) when you substitute a value of \(x\) greater than or equal to \(0\). For example, when you fill in \(x=10\), then you get \(-14 \le -24\) and this is a false statement. Any other value of \(x\) greater than or equal to \(0\) may be used too, and you still get a false statement.
  • From these two numeric examples follows that solutions \(x\) of \(-x -4 \le -2x -4\) must satisfy \(x \le 0\).
    The points where the inequality holds are shown in green in the number line below. An open circle around \(x=0\) 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.