Math Help!?

Question

Solve the inequality

x^4 - x ≤ 0

Answer 1

x^4 - x ≤ 0

factor out x
x (x^3 - 1) ≤ 0

factor, use this formula:
a^3 - b^3 = (a - b) (a^2 + ab + b^2)
x (x - 1) (x^2 + x + 1) ≤ 0

so the roots are 0 and 1

check:

plug in 2
2^4 - 2 ≤ 0
16 - 2 ≤ 0

the statement is false, so x ≤ 1

plug in -1
(-1)^4 - (-1) ≤ 0
1 + 1 ≤ 0

the statement is false, so x ≥ 0


the answer is: 0 ≤ x ≤ 1

Answer 2

4

Answer 3

Solve it like a regular equation first:
x^4 - x = 0
x (x^3 - 1) = 0
x (x - 1) (x^2 + x + 1) = 0

x = 0

x - 1 = 0
x = 1

x^2 + x + 1 = 0
No real solution

Now plot the real solutions on a number line and examine the intervals (Y means the inequality holds, N means it does not hold):
...N.........Y........N
<----- 0 ----- 1 ----->

So the solution set is:
[0, 1]

Answer 4

factor out x
split
solve
x<=1 and x>=0

Answer 5

x^4 - x ≤0
x (x^3 - 1) ≤ 0
x ≤ 0 or x^3 -1 ≤ 0
x^3 ≤ 1
x ≤ 1

therefor, it's 0 or 1