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.