\(x \ge -3\)


We follow the following roadmap:

• Get started with the corresponding equation \[-4x + 8 = 5x {\,+\,}35\]
• Solve this equation:
  
  • Get the terms with \(x\) on the left-hand side of the equation (by adding \(-5x\) on both sides):
    \(-4x + 8 - 5x = 5x {\,+\,}35 - 5x\), which simplifies to \(-9x +8 = 35\).
  • Then move the terms without \(x\) to the right (by adding \(-8\) both sides):
    \(-9x +8 - 8 = 35 - 8\), which simplifies to \(-9x = 27\).
    
    • Next, divide the left- and right-hand side by the coefficient of \(x\) (which is here \(-9\)); this gives \(x = \;\frac{27}{-9}\).
  • 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 \(-4x + 8 \le 5x {\,+\,}35\) when you substitute a value of \(x\) less than or equal to \(-3\). For example, when you fill in \(x=-10\), then you get \(48 \le -15\) 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 \(-4x + 8 \le 5x {\,+\,}35\) when you substitute a value of \(x\) greater than or equal to \(-3\). For example, when you fill in \(x=10\), then you get \(-32 \le 85\) 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 \(-4x + 8 \le 5x {\,+\,}35\) 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.