Question

How do you solve and graph $ - x - 4y > 3?$

Answer 1

Hint: To solve inequality consider it as normal equation and solve for its points by putting random values of independent variable to get value of dependent variable and collecting some points, draw its graph as a normal graph should be drawn and then check for any point on the graph (not the point on line) and if it satisfies the inequality then shade whole portion on which the point lies and also if inequality is not saying equals to then draw the dotted graph of the equation.

Complete step by step solution:
In order to solve and graph the given inequality $ - x - 4y > 3$, we will first simplify this inequality by making the coefficients of variables positive as follows
$
   \Rightarrow - x - 4y > 3 \\
   \Rightarrow - (x + 4y) > 3 \\
 $
Multiplying both sides with $ - 1$, we will get
$
   \Rightarrow - 1 \times - (x + 4y) < - 1 \times 3 \\
   \Rightarrow x + 4y < - 3 \\
 $
Now considering it as a normal equation by removing the inequality sign, in order to find the points to plot the respective graph
$ \Rightarrow x + 4y = - 3$
We can write it as
$ \Rightarrow y = \dfrac{{ - 3 - x}}{4}$
Collecting coordinate of points in the table below:

$x$	$y = \dfrac{{ - 3 - x}}{4}$	Coordinate $(x,\;y)$
$1$	$ - 1$	$(1,\; - 1)$
$ - 3$	$0$	$( - 3,\;0)$
$5$	$ - 2$	$(5,\; - 2)$

So, we got some points $(1,\; - 1),\;( - 3,\;0),\;(5,\; - 2)$ plotting these points on a graph and making a line from it, as follows

Now coming to the inequality,
$ \Rightarrow x + 4y < - 3$
And checking if $(0,\;0)$ is satisfying it or not,
$
   \Rightarrow 0 + 4 \times 0 < - 3 \\
   \Rightarrow 0 < - 3 \\
 $
Since $(0,\;0)$ do not hold good for the given inequation, therefore we will shade portion opposite to $(0,\;0)$ and also since the inequality says less than not less than equals to, so we will draw dotted line

This is the required graph for the inequality.

Note: Inequality sign inverted when we have multiplied $ - 1$ to the equation, let us understand it with example the inequality $2 < 3$ but when we multiply it with $ - 1$ that is $ - 1 \times 2 > - 1 \times 3 \Rightarrow - 2 > - 3$ the inequality changes.