$x \le -3$


We follow the following roadmap:

• Get started with the corresponding equation $$3x -2 = 4x {\,+\,}1$$
• Solve this equation:
  
  • Get the terms with $x$ on the left-hand side of the equation (by adding $-4x$ on both sides):
    $3x -2 - 4x = 4x {\,+\,}1 - 4x$, which simplifies to $-x -2 = 1$.
  • Then move the terms without $x$ to the right (by adding $2$ both sides):
    $-x -2 +2 = 1 +2$, which simplifies to $-x = 3$.
    
    • Next, divide the left- and right-hand side by the coefficient of $x$ (which is here $-1$); this gives $x = \;\frac{3}{-1}$.
  • 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 $3x -2 \ge 4x {\,+\,}1$ when you substitute a value of $x$ less than or equal to $-3$. For example, when you fill in $x=-10$, then you get $-32 \ge -39$ and this is a true statement. Any other value of $x$ less than or equal to $-3$ may be used too, and you still get a true statement.
  • Then calculate the left- and right-hand sides of the inequality $3x -2 \ge 4x {\,+\,}1$ when you substitute a value of $x$ greater than or equal to $-3$. For example, when you fill in $x=10$, then you get $28 \ge 41$ and this is a false statement. Any other value of $x$ greater than or equal to $-3$ may be used too, and you still get a false statement.
  • From these two numeric examples follows that solutions $x$ of $3x -2 \ge 4x {\,+\,}1$ must satisfy $x \le -3$.
    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.