use analytical and graphical methods to solve the inequality.
x^3+10x^2+31<=-30
(2x/x+1)<=(4/x-3)
To solve the inequality x^3 + 10x^2 + 31 ≤ -30 analytically, we need to move all the terms to one side of the inequality.
x^3 + 10x^2 + 31 + 30 ≤ 0
Combining like terms, we get:
x^3 + 10x^2 + 61 ≤ 0
Now let's solve this inequality graphically. We can plot the graph of the function y = x^3 + 10x^2 + 61 and then determine the regions where the graph is less than or equal to zero.
To plot the graph, we can use a graphing calculator or software, which makes it easier to analyze the shape of the curve. From the graph, we can see the portions of the graph that are below or touching the x-axis. These regions represent where the inequality is satisfied.
Regarding the inequality (2x/x+1) ≤ (4/x-3), we can solve it analytically by using algebraic manipulations. Let's start by multiplying both sides of the inequality by (x+1) and (x-3) to clear the denominators.
(x+1)(x-3)(2x) ≤ (x+1)(x-3)(4)
Simplifying the equation, we get:
2x(x-3) ≤ 4(x+1)
Expanding both sides, we have:
2x^2 - 6x ≤ 4x + 4
Rearranging the equation and simplifying, we get:
2x^2 - 10x - 4 ≤ 0
Now, we can solve this quadratic equation analytically or graphically. To solve it graphically, we can plot the graph of the function y = 2x^2 - 10x - 4 and determine the regions where the graph is less than or equal to zero.
By analyzing the graph, we can find the portions of the graph that are below or touching the x-axis. These regions represent where the inequality (2x/x+1) ≤ (4/x-3) is satisfied.