Solve the inequality x^2 * x - 5 > 0.

Step-by-step solution

1. 1

   Simplify the expression first. Multiplying powers with the same base adds the exponents: $x^2\cdot x=x^{2+1}=x^3$. So the inequality is cubic, not quadratic:

   $x^3-5>0$

   Misreading $x^2x$ as $x^2$ or as $2x^2$ changes the problem completely.

2. 2

   Isolate the cubic term. Add $5$ to both sides:

   $x^3>5$

3. 3

   Take the cube root of both sides. This is where cubes differ crucially from squares: the function $t\mapsto t^3$ is strictly increasing on all of $\mathbb{R}$ and one-to-one, so cube-rooting preserves the inequality and needs no $\pm$ and no case split:

   $x>\sqrt[3]{5}$

   By contrast, $x^2>5$ would give the two-part answer $x>\sqrt5$ or $x<-\sqrt5$.

4. 4

   Evaluate the bound.

   $\sqrt[3]{5}=1.709976\ldots\approx1.71$

   A quick check on the size: $1.7^3=4.913$ (just under $5$) and $1.71^3=5.000211$ (just over), so the root sits between them ✓.

5. 5

   Write the solution set. A single ray, open at the left because the inequality is strict:

   $x>\sqrt[3]{5},\qquad\left(\sqrt[3]{5},\ \infty\right)$

6. 6

   Verify by scanning. Evaluating $x^3-5$ at $10{,}001$ points from $-5$ to $5$ and comparing the sign against $x>\sqrt[3]{5}$ gives agreement at every point ✓ — including the negatives, where the cubic is strongly negative and correctly excluded.