Would this be the answer for this problem 2x^3-x^2 > x
x>1 or 0>x>-1/2
To determine the solution to the inequality 2x^3 - x^2 > x, you can follow these steps:
Step 1: Bring all terms to one side of the inequality to have a quadratic expression on one side, which gives:
2x^3 - x^2 - x > 0
Step 2: Rearrange the terms in descending order of powers of x:
2x^3 - x^2 - x - 0 > 0
Step 3: Factor out common terms if possible. In this case, we can factor out an x from each term:
x(2x^2 - x - 1) > 0
Step 4: Solve the quadratic equation. Here, we have:
2x^2 - x - 1 = 0
To solve this quadratic equation, you can either factor it or use the quadratic formula. Let's use the quadratic formula:
x = [-(-1) ± √((-1)^2 - 4(2)(-1))] / (2(2))
Simplifying this equation further, we get:
x = (1 ± √(9)) / 4
x = (1 ± 3) / 4
This leads to two possible solutions:
x = (1 + 3) / 4 = 4 / 4 = 1
x = (1 - 3) / 4 = - 2 / 4 = - 1 / 2
So, the quadratic equation has roots at x = 1 and x = -1/2.
Step 5: Find the sign of the expression for different intervals of x.
- Take a test point x = 0: Substitute the value of x into the expression and check the sign:
2(0)^3 - (0)^2 - (0) = 0 > 0
- Take a test point x = 1/2: Substituting x = 1/2 into the expression:
2(1/2)^3 - (1/2)^2 - (1/2) = (1/4) - (1/4) - (1/2) = -1/2 < 0
- Take a test point x = 2: Substituting x = 2 into the expression:
2(2)^3 - (2)^2 - (2) = 16 - 4 - 2 = 10 > 0
By analyzing the sign of the expression for different intervals, we can determine the intervals where the inequality holds true.
Since the expression is positive when x < -1/2 or x > 1, and negative when -1/2 < x < 1, the solution to the inequality 2x^3 - x^2 > x is given by:
x > 1 or -1/2 < x < 1
Therefore, your answer is partially correct. The correct solution is x > 1 or -1/2 < x < 1.