\(x \lt 5\)


We follow the following roadmap:

• Get started with the corresponding equation \[-5x + 1 = 5x -49\]
• Solve this equation:
  
  • Get the terms with \(x\) on the left-hand side of the equation (by adding \(-5x\) on both sides):
    \(-5x + 1 - 5x = 5x -49 - 5x\), which simplifies to \(-10x +1 = -49\).
  • Then move the terms without \(x\) to the right (by adding \(-1\) both sides):
    \(-10x +1 - 1 = -49 - 1\), which simplifies to \(-10x = -50\).
    
    • Next, divide the left- and right-hand side by the coefficient of \(x\) (which is here \(-10\)); this gives \(x = \;\frac{-50}{-10}\).
  • So, the solution of the equation is \(x = {5}\).

• Find out whether the solutions are on the number line to the left or to the right of \(5\).
  • First calculate the left- and right-hand sides of the inequality \(-5x + 1 \gt 5x -49\) when you substitute a value of \(x\) less than \(5\). For example, when you fill in \(x=-10\), then you get \(51 \gt -99\) and this is a true statement. Any other value of \(x\) less than \(5\) may be used too, and you still get a true statement.
  • Then calculate the left- and right-hand sides of the inequality \(-5x + 1 \gt 5x -49\) when you substitute a value of \(x\) greater than \(5\). For example, when you fill in \(x=10\), then you get \(-49 \gt 1\) and this is a false statement. Any other value of \(x\) greater than \(5\) may be used too, and you still get a false statement.
  • From these two numeric examples follows that solutions \(x\) of \(-5x + 1 \gt 5x -49\) must satisfy \(x \lt 5\).
    The points where the inequality holds are shown in green in the number line below. An open circle around \(x=5\) 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.