1/x is less than or equal to 4/x^3
1/x ≤ 4/x^3
assuming x≠0
multiply both sides by x^3
For x>0 (=> x^3>0)
x²≤4 => x≤2
=> 0<x≤2
For x<0 (=> x^3<0), we need to reverse the direction of the inequality:
x²≥4 => x≤-2
=> -∞<x≤-2
So the final answer is:
0<x≤2 or -∞<x≤-2
In interval notations:
(-∞,-2]∪(0,2]
I got that same interval notation. Except I subtracted the 4/x^3 to the other side and ended up with (x+2)(x-2)/x^3 less than or equal to 0. Then I drew one of those. Line grAphs Nd put circles at -2,2, and an undefined at 0. Then figured out where the x's are negative. Would that be right?
ignoring the ≤ for the time being
1/x = 4/x^3
x^3 = 4x
x^3-4x=0
x(x^2-4) = 0
x(x+2)(x-2) = 0
so critical values are x = 0 and x = 2 , x = -2
4 regions to consider on the number line
1. x < -2 , let's say x = -4
Is 1/-4 ≤ 4/-64 ? YES
2. between -2 and 0 , let say x = -1
is 1/-1 ≤ 4/-1 , NO
3. between 0 and 2 , lets say x = 1.5
is .666 ≤ 1/1.185 ? , YES
4. x > 2 , lets say x = 4
is 1/4 ≤ 4/64 , NO
so x ≤ -2 or 0 < x ≤ 2
(notice x=0 is excluded, since we cannot divide by zero)
Wolfram illustrates the solution rather nicely
http://www.wolframalpha.com/input/?i=1%2Fx+%3D+4%2Fx%5E3
To solve the inequality 1/x ≤ 4/x^3, we first need to find the values of x that satisfy this inequality.
Step 1: Start by multiplying both sides of the inequality by x^3 to eliminate the denominators:
1/x * x^3 ≤ 4/x^3 * x^3
x^2 ≤ 4
Step 2: Rearrange the inequality:
x^2 - 4 ≤ 0
Step 3: Factor the left side of the inequality:
(x + 2)(x - 2) ≤ 0
Step 4: Set each factor equal to zero and solve for x:
x + 2 = 0 --> x = -2
x - 2 = 0 --> x = 2
Step 5: On a number line, plot the solutions -2 and 2:
-----(-2)------(2)-----
Step 6: Choose a test value from each interval to determine whether it satisfies the inequality. For example, chose x = -3, and x = 0:
For x = -3, plug it into the inequality: (-3 + 2)(-3 - 2) ≤ 0
(-1)(-5) ≤ 0
5 ≤ 0 (False)
For x = 0, plug it into the inequality: (0 + 2)(0 - 2) ≤ 0
(2)(-2) ≤ 0
-4 ≤ 0 (True)
Step 7: Analyze the results on the number line:
- The inequality is true when x is between -2 and 2.
- The inequality is false when x is less than -2 or greater than 2.
Therefore, the solution to the inequality 1/x ≤ 4/x^3 is x ∈ (-∞, -2] ∪ [2, +∞).