use analytical and graphical methods to solve the inequality.
x^3+10x^2+31¡Ü-30
(2x/x+1)¡Ü(4/x-3)
To solve the first inequality, x^3 + 10x^2 + 31 ≤ -30, we can use both analytical and graphical methods.
Analytical method:
1. Start by subtracting -30 from both sides of the inequality:
x^3 + 10x^2 + 31 - (-30) ≤ -30 - (-30)
x^3 + 10x^2 + 31 + 30 ≤ 0
x^3 + 10x^2 + 61 ≤ 0
2. Next, we need to determine the critical points where the inequality might change. To find these points, we set the left side of the inequality equal to zero:
x^3 + 10x^2 + 61 = 0
However, this equation does not factor easily. To find the critical points, use numerical methods or a graphing calculator.
By analyzing the graph of the equation or using numerical methods, we find that there are no real solutions for the equation x^3 + 10x^2 + 61 = 0. Therefore, there are no critical points.
3. Since there are no critical points, we can determine the sign of the expression x^3 + 10x^2 + 61 for any value of x. We can observe that the expression is always positive.
Therefore, x^3 + 10x^2 + 61 ≤ 0 has no solutions.
Graphical method:
To solve the inequality graphically, plot the graph of the function f(x) = x^3 + 10x^2 + 31. Then, shade the region below the x-axis since we are looking for the values of x where f(x) ≤ -30.
Upon plotting the graph, we can see that the curve lies entirely above the x-axis. Therefore, there are no values of x that satisfy the inequality x^3 + 10x^2 + 31 ≤ -30.
For the second inequality (2x/(x + 1) ≤ 4/(x - 3)), we can solve it using an analytical method:
1. Begin by multiplying both sides of the inequality by (x + 1)(x - 3) to eliminate the denominators:
(2x/(x + 1)) * (x + 1)(x - 3) ≤ (4/(x - 3)) * (x + 1)(x - 3)
2x(x - 3) ≤ 4(x + 1)
2x^2 - 6x ≤ 4x + 4
2. Rearrange the inequality to get all terms on one side:
2x^2 - 6x - 4x - 4 ≤ 0
2x^2 - 10x - 4 ≤ 0
3. Simplify the quadratic expression if possible, or use the quadratic formula to find the solutions. In this case, we have a simplified form.
4. Now, we need to determine the critical points or break points. To find these points, set the expression inside the inequality equal to zero:
2x^2 - 10x - 4 = 0
Using the quadratic formula:
x = (-(-10) ± √((-10)^2 - 4(2)(-4))) / (2(2))
x = (10 ± √(100 + 32)) / 4
x = (10 ± √132) / 4
Simplifying the radicals:
x = (10 ± 2√33) / 4
x = (5 ± √33) / 2
Therefore, the critical points or break points are x = (5 + √33) / 2 and x = (5 - √33) / 2.
5. To determine the sign of the expression 2x^2 - 10x - 4, we can analyze the intervals defined by the critical points (5 + √33) / 2 and (5 - √33) / 2, by testing a value from each interval and evaluating the expression.
For example, testing x = 0, we have:
2(0)^2 - 10(0) - 4 = -4
Since the result is negative, the interval to the left of (5 - √33) / 2 will satisfy the inequality.
6. Now, we can write the solution to the inequality:
x ≤ (5 - √33) / 2
Alternatively, to solve this inequality using a graphical method, plot the graph of the functions f(x) = 2x/(x + 1) and g(x) = 4/(x - 3). The solution will correspond to the values of x for which f(x) ≤ g(x).
By analyzing the graph, we can observe that f(x) ≤ g(x) when x ≤ (5 - √33) / 2.
I hope this helps! Let me know if you have any further questions.