Question

Can i get hep with this?

Answer 1

The solution set of the inequality in interval notation is: x ∈ [- 3, - 12 / 7)

How to find the solution of a inequation involving power function with rational exponents

Herein we have an inequality with power functions of rational exponents, whose solution can be found by using algebra properties. The detail procedure is presented below:

(2 / 3) · x$^{-\frac{1}{3} }$ · (x + 3)$^{\frac{1}{2} }$ + (1 / 2) · x$^{\frac{2}{3} }$ · (x + 3)$^{-\frac{1}{2} }$ < 0                  Given

(2 /3) · (x + 3) + (1 / 2) · x < 0                                               Compatibility with multiplication / Multiplication of power with equal base / Associative and modulative properties

(7 / 6) · x + 2 < 0                                                                  Distributive property / Definition of addition

(7 / 6) · x < - 2                                                                      Compatibility with addition / Existence of additive inverse / Modulative property

x < - 12 / 7                                                                            Compatibility with multiplication / Existence of multiplicative inverse / Associative and modulative properties / Result

But x + 3 ≥ 0 must be greater than 0 to bring out a real solution, then the solution set of the inequality in interval notation is: x ∈ [- 3, - 12 / 7)

To learn more on inequalities: brainly.com/question/20383699

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