Use analytical and graphical methods to solve the inequality:

x©ø+10x©÷+31x¡-30

(2x)/(x+1)¡(4)/(x-3)

To solve the inequality, let's start by simplifying the expression:

x² + 10x + 31x - 30

Combining like terms, we get:

x² + 41x - 30

Now, we will analyze the inequality and graph it:

(2x)/(x+1) - (4)/(x-3) > 0

Let's find the critical points by setting the numerator and denominator of each fraction equal to zero:

For the first fraction, (2x)/(x+1), the numerator is zero when 2x = 0, so x = 0. The denominator is zero when x + 1 = 0, so x = -1.

For the second fraction, (4)/(x-3), the numerator is zero when 4 = 0, which is not possible. The denominator is zero when x - 3 = 0, so x = 3.

Therefore, the critical points are x = -1, 0, and 3.

Next, we will create a number line and plot these critical points on it:

-∞ -1 0 3 +∞

Now, we need to test intervals between the critical points to determine the solution. Let's test two intervals: (-∞, -1), (-1, 0), (0, 3), and (3, +∞).

For (-∞, -1), let's pick a test point, such as x = -2:

(2(-2))/(-2+1) - (4)/(-2-3) = 4/(-1) - 4/(-5) = -4 + 4/5 = 0.2

Since 0.2 is greater than zero, this interval satisfies the inequality.

For (-1, 0), let's pick a test point, such as x = -0.5:

(2(-0.5))/(-0.5+1) - (4)/(-0.5-3) = -1/1.5 - 4/-3.5 = -0.67 + 1.14 = 0.47

Since 0.47 is greater than zero, this interval satisfies the inequality.

For (0, 3), let's pick a test point, such as x = 1:

(2(1))/(1+1) - (4)/(1-3) = 2/2 - 4/-2 = 1 + 2 = 3

Since 3 is greater than zero, this interval satisfies the inequality.

For (3, +∞), let's pick a test point, such as x = 4:

(2(4))/(4+1) - (4)/(4-3) = 8/5 - 4/1 = 1.6 - 4 = -2.4

Since -2.4 is less than zero, this interval does not satisfy the inequality.

To summarize the solution, the inequality is satisfied for all values of x less than -1 and between -1 and 0, as well as between 0 and 3. Therefore, the solution is x < -1 or -1 < x < 0 or 0 < x < 3.