Question

How do you graph the inequality $2x-3y\le 6$?

Answer 1

Hint: We start solving the problem by finding the point on y-axis for which the equality holds for the given inequality by using the fact that the value of x-coordinate on the y-axis is zero. We then find the point on x-axis for which the equality holds for by using the fact that the value of y-coordinate on the x-axis is zero. We then represent these two points on the plot and then join those points to get the required plot and then map the area at which the inequality holds true.

Complete step-by-step answer:
According to the problem, we are asked to plot the given inequality $2x-3y\le 6$.
So, let us find the points on x-axis and y-axis when the equality holds for given $2x-3y\le 6$.
Now, let us find the point on the y-axis for which the equality holds for $2x-3y\le 6$. We know that the value of x-coordinate on the y-axis is zero. Let us substitute $x=0$ in the given equation to find the intersection point.
So, we get $2\left( 0 \right)-3y=6$.
$\Rightarrow 0-3y=6$.
$\Rightarrow -3y=6$.
$\Rightarrow y=\dfrac{6}{-3}$.
$\Rightarrow y=-2$.
So, the point that lies on the y-axis when equality holds for $2x-3y\le 6$ is $A\left( 0,-2 \right)$.
Now, let us find the point on the x-axis for which the equality holds for $2x-3y\le 6$. We know that the value of y-coordinate on the x-axis is zero. Let us substitute $y=0$ in the given equation to find the intersection point.
So, we get $2x-3\left( 0 \right)=6$.
$\Rightarrow 2x-0=6$.
$\Rightarrow 2x=6$.
$\Rightarrow x=\dfrac{6}{2}$.
$\Rightarrow x=3$.
So, the point that lies on the x-axis when equality holds for $2x-3y\le 6$ is $B\left( 3,0 \right)$.
Now, let us plot the given line by representing and joining the obtained two points and then mapping the area at which the inequality holds true on the graph which is as shown below:

Note: Whenever we get this type of problems, we first try to find the points at which the given curve intersects both the axes. We can also plot the given equation by assuming the values for x to find the points of y which will help to plot on the graph. Similarly, we can expect problems to plot the inequality $6x-2y\ge 12$ on the graph.