Question

How do you solve \[3x\le 2x-3\]?

Answer 1

Hint: Rearrange the expression by taking the terms containing the variable x to the L.H.S. and leaving the constant terms to the R.H.S. Now, simplify the L.H.S. by simple addition or subtraction, whichever needed, and make the coefficient of x equal to 1. Leave the inequality sign as it is. Draw a line x = -3 and consider the suitable part of the graph according to the simplified inequality obtained.

Complete step by step solution:
Here, we have been provided with the inequality \[3x\le 2x-3\] and we are asked to solve it. So, let us solve it algebraically first and then see how it looks graphically.
Now, rearranging the inequality by taking the terms containing the variable x to the L.H.S. and leaving the constant terms in the R.H.S., we get,
\[\Rightarrow 3x\le 2x-3\]
\[\Rightarrow x\le -3\] ……………………... (1)
Here, you may see that we have not changed the direction of inequality sign because the direction of inequality sign only changes when we multiply or divide both the sides with a negative number or take reciprocal on both sides.
Now, let us draw the graph of inequality obtained in (1). To do this, first we have to remove the inequality sign and replace it with ‘=’ sign and draw the required line. So, we have,
\[\Rightarrow x=-3\]
Drawing the line x = -3, we get,

Now, considering inequation (1), i.e., \[x\le -3\], here we can clearly see that we have to select that part of the graph in which x will be less than or equal to -3. So, in the above graph we have to select the left side of the graph. Therefore, we have,

Note: One may note that there are not many differences in solving and graphing an equality and an inequality. We need the help of equality while drawing the graph. One thing you may note is we have to consider the value of x = -3 in our graph that is why we have used to consider the value of x = -3 in our graph that is why we have used solid line in the graph for x = -3 otherwise we would have used the dashed line. Always remember the rules of reversing the direction of inequality. The direction is only reversed when we take reciprocal or divide and multiply both the sides with a negative number.