Someone help me!!
square root = sqrt
(5x^2-5x-10)sqrt(x+7) + (2x+6)sqrt(x+2) >= x^3+13x^2-6x-28
thank you!
well, looking at the graphs, it looks like -2 <= x <= 2
checking, if we call them
f(x)+g(x) >= h(x), then
f(-2) = 20√5
g(-2) = 0
f(2) = 0
g(2) = 20
h(-2) = 28
h(2) = 20
so,
f(-2)+g(-2) >= h(-2)
f(2) + g(2) >= h(2)
Clearly we need x >= -2 for √(x+2) to be real.
Beyond x=2, h(x) clearly grows faster than f+g, since h is a cubic (x^3) and f+g is basically x^2.5, so for x>2, f+g < h
The graph of (f+g)-h is shown here:
http://www.wolframalpha.com/input/?i=plot+y%3D%285x^2-5x-10%29%E2%88%9A%28x%2B7%29+%2B+%282x%2B6%29%E2%88%9A%28x%2B2%29+,+y%3Dx^3%2B13x^2-6x-28+for+-2+%3C%3D+x+%3C%3D+3
To solve the given inequality, we need to find the values of x that satisfy the inequality. Let's break it down into smaller steps:
Step 1: Simplify the given inequality.
We start by expanding both sides of the inequality:
sqrt(5x^2 - 5x - 10) * sqrt(x + 7) + sqrt(2x + 6) * sqrt(x + 2) >= x^3 + 13x^2 - 6x - 28
Step 2: Further simplify the equation.
Combine the square roots on both sides of the inequality by multiplying them:
sqrt[(5x^2 - 5x - 10)(x + 7)] + sqrt[(2x + 6)(x + 2)] >= x^3 + 13x^2 - 6x - 28
Step 3: Expand the products within the square roots:
sqrt[5x^3 + 40x^2 + 85x + 70] + sqrt[2x^2 + 10x + 12] >= x^3 + 13x^2 - 6x - 28
Step 4: Move the terms containing the square roots to one side of the inequality, and all the other terms to the opposite side:
sqrt[5x^3 + 40x^2 + 85x + 70] + sqrt[2x^2 + 10x + 12] - x^3 - 13x^2 + 6x + 28 >= 0
Step 5: Simplify the equation further if possible.
Rearrange the terms and combine like terms:
sqrt[5x^3 + 40x^2 + 85x + 70 - x^6 - 26x^5 + 13x^4 - 169x^3 + 78x^2 + 28x^2 + 364x - 12x^2 - 56x - 364] >= 0
Step 6: Combine similar terms within the square root:
sqrt[-x^6 - 26x^5 + 13x^4 - 169x^3 + 90x^2 + 336x - 364] >= 0
Step 7: Determine the values of x that make the inequality true.
To find the valid x-values, we need to look at the square root expression on the left side of the inequality. The square root of any value must be greater than or equal to zero.
So, to solve the inequality, we need to solve the expression within the square root to be greater than or equal to zero:
-x^6 - 26x^5 + 13x^4 - 169x^3 + 90x^2 + 336x - 364 >= 0
You can solve this equation by either factoring it or using a graphing calculator. The solutions to this inequality will determine the range of x-values that satisfy the original inequality.
I hope this explanation helps you in solving the given inequality.