Question

How do you graph the inequality \[2x + y \leqslant 6\] on the coordinate plane ?

Answer 1

Hint:First we need to draw the graph of the equation \[2x + y = 6\]. We use intercept form to draw the graph. That is we find the coordinate of the given equation lying on the line of x- axis, we can find this by substituting the value of ‘y’ is equal to zero (x-intercept). Similarly we can find the coordinate of the equation lying on the line of y- axis, we can find this by substituting the value of ‘x’ equal to zero (y-intercept). After drawing the graph we can check in which region the inequality satisfies.

Complete step by step answer:
Given, \[2x + y \leqslant 6\]. Now consider \[2x + y = 6\]. To find the x-intercept. That is the value of ‘x’ at\[y = 0\]. Substituting this in the given equation. We have,
\[2x + 0 = 6\]
\[\Rightarrow 2x = 6\]
Divide by 2 on both side,
\[x = \dfrac{6}{2}\]
\[ \Rightarrow x = 3\]
Thus we have a coordinate of the equation which lies on the line of x-axis. The coordinate is \[(3,0)\]. To find the y-intercept. That is the value of ‘y’ at \[x = 0\]. Substituting this in the given equation we have,
\[2(0) + y = 6\]
\[ \Rightarrow y = 6\]
Thus we have a coordinate of the equation which lies on the line of y-axis. The coordinate is \[(0,6)\]. Thus we have the coordinates \[(3,0)\] and \[(0,6)\]. Let’s plot a graph for these coordinates. We take scale x-axis= 1 unit = 1 units and y-axis= 1 unit = 1 units.

We expanded the point touching the intercepts. We took a coordinate above and below the equation of line (see in above graph).
That is \[(x,y) = (2,3)\] and now put it in the inequality,
 \[2(2) + (3) \leqslant 6\]
\[4 + 3 \leqslant 6\]
\[ \Rightarrow 7 \leqslant 6\]. Which is wrong.
Now take a coordinate below the equation of line,
That is \[(x,y) = (1,1)\]
\[2(1) + 1 \leqslant 6\]
\[2 + 1 \leqslant 6\]
\[ \Rightarrow 3 \leqslant 6\]. Which is true.
In the above graph the shaded region is the solution of the given inequality.

Note:If we take any coordinate point below the line of the graph, the inequality satisfies. Also if we take a point on the line, the inequality will be satisfied. A graph shows the relation between two variable quantities, it contains two axes perpendicular to each other namely the x-axis and the y-axis. Each variable is measured along one of the axes. In the question, we are given one linear equation containing two variables namely x and y, x is measured along the x-axis and y is measured along the y-axis while tracing the given equations.