Question

How do you sketch the solution for the system of inequalities \[y\le -x-2\] and \[y\ge -5x+2\]?

Answer 1

Hint: In this problem, we have to sketch for the system of given inequalities. We know that, to sketch this problem, we have to find the x-intercept and the y-intercept of the two given inequalities. We can first change the given inequalities to equations, so that we can solve and find the x-intercept and the y-intercept to sketch the problem.

Complete step by step answer:
We know that the given inequalities are,
\[y\le -x-2\]
\[y\ge -5x+2\]
Now we can change the both inequalities to equations so that we can solve for the intercepts, we get
\[y=-x-2\]….. (1)
\[y=-5x+2\]….. (2)
Now we can find the intercepts from the above two equations.
We know that at x-intercept the value of y is 0 and at y-intercept the value of x is 0.
We can take the equation (1),
At y-intercept, x = 0, then from equation (1)
\[\begin{align}
  & \Rightarrow y=-0-2 \\
 & \Rightarrow y=-2 \\
\end{align}\]
The y-intercept of equation (1) is \[\left( 0,-2 \right)\]
At x-intercept, y = 0, then from equation (1)
\[\begin{align}
  & \Rightarrow 0=-x-2 \\
 & \Rightarrow x=-2 \\
\end{align}\]
The x-intercept of equation (1) is \[\left( -2,0 \right)\] .
We can take the equation (2),
At y-intercept, x = 0, then from equation (2)
\[\begin{align}
  & \Rightarrow y=5\left( 0 \right)-2 \\
 & \Rightarrow y=2 \\
\end{align}\]
The y-intercept of equation (2) is \[\left( 0,2 \right)\]
At x-intercept, y = 0, then from equation (2)
\[\begin{align}
  & \Rightarrow 0=5x-2 \\
 & \Rightarrow x=\dfrac{2}{5} \\
\end{align}\]
The x-intercept of equation (2) is \[\left( \dfrac{2}{5},0 \right)\] .
Now we can plot the graph using the points,
The x-intercept of equation (1) is \[\left( -2,0 \right)\] .
The y-intercept of equation (1) is \[\left( 0,-2 \right)\]
The x-intercept of equation (2) is \[\left( \dfrac{2}{5},0 \right)\] .
The y-intercept of equation (2) is \[\left( 0,2 \right)\]

Note:
We should know that the solution set of equation (1), is the area below the line, as the given inequality is less than symbol, similarly the solution set of equation (2) is the area above the line as the given inequality is greater than symbol. The combined solution set is the area commonly shared by two given inequalities.