Solve the following inequalities.
\left(2x-3\right)^{-1/4} \lt \sqrt2
The function \left(\left(2x-3\right)^{-\frac{1}{4}}\right) is defined for x\geqslant\dfrac{3}{2}.
Applying the function x^{-4} (which is decreasing) on both sides of the inequality:
\left( \left(2x-3\right)^{-\frac{1}{4}} \right)^{-4} \gt \left(\sqrt{2}\right)^{-4}
Which is equivalent to:
2x-3 \gt \dfrac{1}{\left(\sqrt{2}\right)^{4}}
2x-3 \gt \dfrac{1}{4}
2x \gt \dfrac{13}{8}
x \gt \dfrac{13}{8}
The solution is included in the definition domain.
The solution of the inequality is \left( \dfrac{13}{8},\infty \right).
\left(4x+1\right)^{-1/2} \lt 3
The function \left(\left(4x+1\right)^{-\frac{1}{2}}\right) is defined for x\geqslant-\dfrac{1}{4}.
Raise both sides to the power:
\left( \left(4x+1\right)^{-\frac{1}{2}} \right)^{-2} \gt 3^{-2}
Which is equivalent to:
4x+1 \gt \dfrac{1}{9}
4x \gt \dfrac{1}{9}-1
4x \gt -\dfrac{8}{9}
x \gt -\dfrac{2}{9}
The solution is included in the definition domain.
The solution of the inequality is \left( -\dfrac{2}{9},\infty \right).
\left(3x-11\right)^{3/2} \leqslant 8
The function \left(3x-11\right)^{\frac{3}{2}} is defined for x\geqslant\dfrac{11}{3}.
Raise both sides to the \left( \dfrac{2}{3} \right) power:
\left( \left(3x-11\right)^{\frac{3}{2}} \right)^{\frac{2}{3}} \leqslant \left(8\right)^{\frac{2}{3}}
Which is equivalent to:
3x-11 \leqslant 8^{\frac{2}{3}}
3x-11 \leqslant4
3x \leqslant15
x \leqslant5
The intersection of the solution and the definition domain is:
\dfrac{11}{3}\leqslant x \leqslant 5
The solution of the inequality is \left[ \dfrac{11}{3},5 \right].
\left(2x+5\right)^{-1/3} \gt 2
The function \left(2x+5\right)^{-\frac{1}{3}} is defined for x \geq -\dfrac{5}{2}.
Raise both sides to the \left(-3\right) power:
\left( \left(2x+5\right)^{-\frac{1}{3}} \right)^{-3} \lt 2^{-3}
Which is equivalent to:
2x+5 \lt \dfrac{1}{8}
2x \lt \dfrac{1}{8}-5
2x \lt -\dfrac{39}{8}
x \lt -\dfrac{39}{16}
The intersection of the solution and the definition domain is:
\dfrac{-5}{2}\leqslant x \leqslant -\dfrac{39}{16}
The solution of the inequality is \left[-\dfrac{5}{2} ,-\dfrac{39}{16}\right).
\sqrt{x-2} \lt 3
The function \sqrt{x-2} is defined for x \geq 2.
Raise both sides to the 2nd power:
\left( \sqrt{x-2} \right)^{2} \lt 3^{2}
Which is equivalent to:
x-2 \lt 9
x \lt 11
The intersection of the solution and the definition domain is:
2\leqslant x \lt 11
The solution of the inequality is \left[2{,}11 \right).
\sqrt{2x+5} \geqslant 4
The function \sqrt{2x+5} is defined for x\geqslant-\dfrac{5}{2}.
Raise both sides to the 2nd power:
\left( \sqrt{2x+5} \right)^{2} \geqslant \left(4\right)^{2}
Which is equivalent to:
2x+5 \geqslant 16
2x \geqslant 11
x\geqslant\dfrac{11}{2}
Because the solution is included in the definition domain:
x\geqslant\dfrac{11}{2}
The solution of the inequality is \left[\dfrac{11}{2},\infty \right).
\sqrt{3x-8} \leqslant 5
The function \sqrt{3x-8} is defined for x\geqslant\dfrac{8}{3}.
Raise both sides to the 2nd power:
\left( \sqrt{3x-8} \right)^{2} \leqslant\left(5\right)^{2}
Which is equivalent to:
3x-8 \leqslant 25
3x \leqslant 33
x\leqslant11
The intersection of the solution and the definition domain is:
\dfrac{8}{3}\leqslant x\leqslant11
The solution of the inequality is \left[\dfrac{8}{3},11 \right].