Question

How do you graph the inequality \[3x - y \leqslant 6\] ?

Answer 1

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

We expanded the point touching the intercepts.
The shaded region is the solution of the given inequality.

Note: If we take any coordinate point on the line of the graph, the inequality satisfies. Also if we take a point on the right side of the graph line (unshaded region), the inequality will not 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.