Question

Solve the following linear inequality to find the value of $x$
$x - 2 \leqslant \dfrac{{5x + 8}}{3}$

Answer 1

Hint: This is a problem with inequality. Solve it as we do for equation but one more rule changing the sign of inequality while multiplying on both sides with negative numbers.

Complete step by step answer:
As we solve the equation with equality sign in between, we do the same here. We will rearrange the terms of the equation on both sides and then only be careful while multiplying with negative numbers on both sides. The inequality sign will reverse on doing so. So, the steps are as follows:
$
   \Rightarrow x - 2 \leqslant \dfrac{{5x + 8}}{3} \\
   \Rightarrow 3(x - 2) \leqslant 5x + 8 \\
   \Rightarrow 3x - 6 - 5x \leqslant + 8 \\
   \Rightarrow - 2x \leqslant 8 + 6 \\
   \Rightarrow - 2x \leqslant 14 \\
$
Now, we will divide both sides with (-2). The sign of equality will change to greater than equal ($ \geqslant $).
$ \Rightarrow x \geqslant - 7$. This is our solution. So, we see that we started off with $ \leqslant $ and ended with $ \geqslant $. Our focus is with solving the problem by finding a value of $x$. Here $x$ can be any value greater than or equal to -7.

Note: Bringing the variables on one side of the equation is just convention and makes it more legible. Instead we can come to the same solution if we gather the variables to the right hand side of the inequality.