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# How do you solve the inequality ${x^3} - {x^2} - 6x > 0$?

Last updated date: 16th Jun 2024
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Hint: An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. Here we need to solve for ‘x’ which is a variable. Solving the given inequality is very like solving equations and we do most of the same thing but we must pay attention to the direction of inequality$( \leqslant , > )$. We can take ‘x’ common and we will have a quadratic equation and we can solve it easily.

Complete step by step solution:
We find the factor of ${x^3} - {x^2} - 6x$ and we evaluate for each term.
Taking ‘x’ common we have,
$x({x^2} - x - 6) > 0$
We can split the middle term in parentheses,
$x({x^2} - 3x + 2x - 6) > 0$
$x\left( {x(x - 3) + 2(x - 3)} \right) > 0$
$x(x - 3)(x + 2) > 0$
Since we have greater than sign it means that $x \ne 0$.
If $x > 0$ then other two factors must be greater than zero. That is
$(x - 3)(x + 2) > 0$
That is $x > 3$. (That is if we have$x < 3$ then will have $(x - 3)$ is negative and since $(x + 2)$ is positive then the product becomes negative which is contradiction to the fact $x > 0$)
If $x < 0$ then $(x - 3)$ will be negative then $(x + 2)$ must be positive. So that the product of $x < 0$,$(x - 3)$ and $(x + 2)$ will be greater than zero.
Thus $(x + 2) > 0$
$x > - 2$

Hence the solution of ${x^3} - {x^2} - 6x > 0$ are $x > 3$ or $- 2 < x < 0$.

Note: We know that $a \ne b$ says that ‘a’ is not equal to ‘b’. $a > b$ means that ‘a’ is less than ‘b’. $a < b$ means that ‘a’ is greater than ‘b’. These two are known as strict inequality. $a \geqslant b$ means that ‘a’ is less than or equal to ‘b’. $a \leqslant b$ means that ‘a’ is greater than or equal to ‘b’.
The direction of inequality do not change in these cases:
i) Add or subtract a number from both sides.
ii) Multiply or divide both sides by a positive number.
iii) Simplify a side.
iv) The direction of the inequality change in these cases:
v) Multiply or divide both sides by a negative number.
vi) Swapping left and right hand sides.