Question

Need the steps to solve this inequality:

Answer 1

\[x^4-x \le 0\]

Answer 2

hint:
we have to factorize the binomial at the left side

Answer 3

for example, at the first step, we can write:
\[\Large x\left( {{x^3} - 1} \right) \leqslant 0\]

Answer 4

now, we have to factorize x^3-1, do you know how to factorize it?

Answer 5

Yes

Answer 6

ok! Then rewrite my inequality above, using your factorization

Answer 7

would it be
(x-1)( x+1)( x -1)?

Answer 8

not exactly, we have this:
\[\Large x\left( {x - 1} \right)\left( {{x^2} + x + 1} \right) \leqslant 0\]

Answer 9

since:
\[\Large {x^3} - 1 = \left( {x - 1} \right)\left( {{x^2} + x + 1} \right)\]

Answer 10

I see

Answer 11

now, we can note that \[\Large {{x^2} + x + 1}\] is always positive

Answer 12

so we have to study the sign of x and x-1 only

Answer 13

for example, please solve this inequality:
\[\Large x - 1 \geqslant 0\]

Answer 14

\[x \ge 1\]

Answer 15

correct! So we have this drawing:

Answer 16

a continuous line stands for positivity, whereas a dashed line stands for negativity

Answer 17

so we have the subsequent drawing: