Question

How do you graph the inequality $y\le 4x-1$?

Answer 1

In this problem we need to draw off the given inequality. For this we need to first draw the graph of the equality. In this problem we have the inequality $y\le 4x-1$. Now the equality of the above inequality will become as $y=4x-1$. Now we will draw the graph of the equation $y=4x-1$ calculating the intercepts of the equation. We know that intercepts are the points where the given line meets the coordinate axis. To calculate intercepts, we will substitute $x=0$, $y=0$ in the given equation and we will calculate the remaining variable value. After calculating the intercept points, we will join them to get the graph of the line $y=4x-1$. But we need to draw the graph of inequality $y\le 4x-1$. In the inequality we have less than symbols so we will shade the area which is under the line $y=4x-1$ and that will be our required graph.

Complete step-by-step solution:
Given inequality $y\le 4x-1$.
Considering the equation $y=4x-1$.
Substituting $x=0$ in the above equation, then we will have
$\begin{aligned} & y=4\left( 0 \right)-1 \\\ & \Rightarrow y=-1 \\\ \end{aligned}$
So, the point $\left( 0,-1 \right)$ is the $x-$intercept.
Substituting $y=0$ in the above equation, then we will get
$\begin{aligned} & 0=4x-1 \\\ & \Rightarrow 4x=1 \\\ & \Rightarrow x=\frac{1}{4} \\\ \end{aligned}$
So, the point $\left( \frac{1}{4},0 \right)$ is the $y-$intercept.
Plotting the two points in a graph paper, then we will get

Joining the two points, then we will get the graph of the line $y=4x-1$ as

Now the graph of the inequality $y\le 4x-1$ will be the area under the line $y=4x-1$. So, we are going to shade the area under the line $y=4x-1$ to show a graph of the inequality $y\le 4x-1$.

Note: In this problem they have asked to draw the inequality $y\le 4x-1$. So, we have shaded the area which is under the line $y=4x-1$. If they asked to draw the inequality $y\ge 4x-1$, then we will shade the area which is above the line $y=4x-1$.