Find the solution set to the following rational inequality, 240/x+8 > 20x/x+1
To solve the rational inequality 240/(x+8) > 20x/(x+1), we need to find the values of x that make the inequality true. Here's how you can do it step by step:
1. Start by multiplying both sides of the inequality by (x+8) and (x+1) to eliminate the denominators:
240(x+1) > 20x(x+8)
This gives us:
240x + 240 > 20x^2 + 160x
2. Rearrange the equation to bring all terms to one side and set it equal to zero:
20x^2 + 160x - 240x - 240 > 0
20x^2 - 80x - 240 > 0
3. Divide the entire inequality by 20 to simplify the equation:
x^2 - 4x - 12 > 0
4. Factorize the quadratic expression:
(x - 6)(x + 2) > 0
5. Now we determine the sign of each factor:
- For (x - 6), it is positive (greater than zero) when x > 6.
- For (x + 2), it is positive (greater than zero) when x > -2.
6. Next, we look for the intervals where the expression (x - 6)(x + 2) is positive:
- When x > 6, both factors are positive.
- When -2 < x < 6, (x - 6) is negative and (x + 2) is positive. However, because the inequality is strict (> instead of ≥), we exclude this interval.
7. Finally, we conclude that the solution set to the given rational inequality is:
x > 6
Therefore, any value of x greater than 6 makes the original inequality, 240/(x+8) > 20x/(x+1), true.