\(x \le 4\)


We follow the following roadmap:

• Get started with the corresponding equation \[5x + 6 = 4x {\,+\,}10\]
• Solve this equation:
  
  • Get the terms with \(x\) on the left-hand side of the equation (by adding \(-4x\) on both sides):
    \(5x + 6 - 4x = 4x {\,+\,}10 - 4x\), which simplifies to \(x +6 = 10\).
  • Then move the terms without \(x\) to the right (by adding \(-6\) both sides):
    \(x +6 - 6 = 10 - 6\), which simplifies to \(x = 4\).
    
  • So, the solution of the equation is \(x = {4}\).

• Find out whether the solutions are on the number line to the left or to the right of \(4\).
  • First calculate the left- and right-hand sides of the inequality \(5x + 6 \le 4x {\,+\,}10\) when you substitute a value of \(x\) less than or equal to \(4\). For example, when you fill in \(x=-10\), then you get \(-44 \le -30\) and this is a true statement. Any other value of \(x\) less than or equal to \(4\) may be used too, and you still get a true statement.
  • Then calculate the left- and right-hand sides of the inequality \(5x + 6 \le 4x {\,+\,}10\) when you substitute a value of \(x\) greater than or equal to \(4\). For example, when you fill in \(x=10\), then you get \(56 \le 50\) and this is a false statement. Any other value of \(x\) greater than or equal to \(4\) may be used too, and you still get a false statement.
  • From these two numeric examples follows that solutions \(x\) of \(5x + 6 \le 4x {\,+\,}10\) must satisfy \(x \le 4\).
    The points where the inequality holds are shown in green in the number line below. An open circle around \(x=4\) 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.