+129840 √5 * (1/5)3x + 6.5 ≤ 0.2x(x + 4) and we can write this as
√5 * (1/5)3x + 6.5 ≤ (1/5)x(x + 4) let's just set this equal and solve for x......
√5 *(1/5)3x + 6.5 = (1/5)x(x + 4) divide both sides by (1/5)3x + 6.5
√5 = (1/5)x(x + 4) / (1/5)3x + 6.5 and we can write this as
√5 = (1/5)[ x (x + 4) - 3x - 6.5] simplify the exponent
√5 = (1/5) [ x^2 + 4x - 3x - 6.5]
√5 = (1/5) [ x^2 + x - 6.5] and we can write
5^(1/2) = (5^-1)^ [ x^2 + x - 6.5] simplify, again
5^(1/2) = (5)^ [-x^2 - x + 6.5] note....now we can solve for the exponents since the bases are equal
(1/2) = -x^2 - x + 6.5 multiply through by 2
1 = -2x^2 - 2x + 13 rearrange
2x^2 + 2x - 12 = 0 divide through by 2
x^2 + x - 6 = 0 factor
(x + 3) ( x - 2) = 0 setting each factor to 0, we have that
x = -3 or x = 2
Going back to the original inequality....our answers will either come from these two intervals (inf, -3] U [2, inf) or from this interval [-3, 2]
We can see if the last interval makes our original problem true by testing 0......if it makes the original problem true.....this is the solution interval
√5 * (1/5)3(0) + 6.5 ≤ 0.20(0 + 4) ????
√5 * (1/5)^6.5 ≤ 1 ??????
≈ 0.000143108... ≤ 1 and this is true....so...the answer is -3 ≤ x ≤ 2
Note: WolframAlpha confirms our solution :
https://www.wolframalpha.com/input/?i=%E2%88%9A5+*+(1%2F5)%5E%5B3x+%2B+6.5%5D%C2%A0%E2%89%A4+0.2%5E%5Bx(x+%2B+4)%5D
