Question

How do you graph \[y<-\dfrac{3}{4}x+2\]?

Answer 1

Hint: This question is based on an inequality equation graph, so first we have to convert the inequality symbol to an equal symbol, to find the boundary line below which the equation lies. After converting inequality to equality, we can find the x and y-intercept, where the boundary line lies and we can plot those points to get the graph.

Complete step-by-step answer:
We know that the given equation is,
\[y<-\dfrac{3}{4}x+2\]
Since, the given equation is a linear inequality equation, to graph the boundary line we have to momentarily change the inequality symbol to equality symbol, that is less than symbol to equal symbol.
Now, we can write this equation as,
\[y=-\dfrac{3}{4}x+2\]…….. (1)
Now we are going to find the x-intercept and y-intercept, where the graph crosses x and y-axis,
We can find y-intercept,
We know that at y-intercept, x=0.
Now we can substitute the value of x in equation (1), we get
\[\begin{align}
  & \Rightarrow y=-\dfrac{3}{4}\left( 0 \right)+2 \\
 & \Rightarrow y=2 \\
\end{align}\]
We got the point at y-intercept \[\left( 0,2 \right)\]
Now we can find x-intercept, y=0.
Now we can substitute the value of y in (1), we get
\[\begin{align}
  & \Rightarrow 0=-\dfrac{3}{4}x+2 \\
 & \Rightarrow x=2\times \dfrac{4}{3} \\
 & \Rightarrow x=\dfrac{8}{3} \\
\end{align}\]
Here the point at x-intercept is in fraction, so we can take x=4 in (1), we get
\[\begin{align}
  & \Rightarrow y=-\dfrac{3}{4}\times 4+2 \\
 & \Rightarrow y=-1 \\
\end{align}\]
Now we got another point \[\left( 4,-1 \right)\]
Therefore, the two points are \[\left( 0,2 \right)\]and \[\left( 4,-1 \right)\].
We also have to note that y is less than x, the area below the line is the graph for \[y<-\dfrac{3}{4}x+2\].
Now we can plot these points in the graph for \[y<-\dfrac{3}{4}x+2\].

Note: It is important to note that, in this given sum, we have an inequality equation, we have to change the less than symbol to equal symbol in order to find the x-intercept and y-intercept, that is the boundary line, below which the equation lies. If the given problem has greater than symbol, we can just remember that the equation lies above the boundary line for that respective equation.