How do you solve #x^3-2x>1#?

Answer 1

#(-1, (1 - sqrt5)/2) and ((1 + sqrt5)/2, +inf)#

#f(x) = x^3 - 2x - 1 > 0# The simplest method is graphing the function f(x), then looking for the parts of the graph that are above the x-axis. graph{x^3 - 2x - 1 [-2.5, 2.5, -1.25, 1.25]} On this graph, the x-intercepts are approximately: x = - 1 , x = - 0.65, and x = 1.60 Answers by intervals: (-1, - 0.65) and (1.60, + inf.)
Note 1 . We can find the exact values of the 3 x-intercepts by solving the equation f(x) = x^3 - 2x - 1 = (x + 1)(x^2 - x - 1) = 0 The quadratic equation (x^2 - x - 1) = 0 gives 2 real roots: x = (1 +- sqrt5)/2. Therefor, the answers are: #(-1, (1 - sqrt5)/2)# and #((1 + sqrt5)/2 , +inf.)# Note 2 . Graphing calculator may give much more accurate values of the three x-intercepts. Note 3. Another method is solving algebraically the inequality by creating a Sign Chart. See algebra books.
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Answer 2

To solve the inequality (x^3 - 2x > 1), we first set the inequality equal to zero to find the critical points:

(x^3 - 2x - 1 > 0).

Next, we find the roots of the equation:

(x^3 - 2x - 1 = 0).

Using numerical methods or a graphing calculator, we find that the roots are approximately -1.32, 0.62, and 1.70.

These roots divide the number line into four intervals: ((-∞, -1.32)), ((-1.32, 0.62)), ((0.62, 1.70)), and ((1.70, ∞)).

We then test a value in each interval in the inequality (x^3 - 2x > 1) to determine the sign of the expression in each interval.

For example, for (x = -2), ( (-2)^3 - 2(-2) > 1), which is true. Therefore, the interval ((-∞, -1.32)) is part of the solution.

For (x = 0), (0^3 - 2(0) > 1), which is false. Therefore, the interval ((-1.32, 0.62)) is not part of the solution.

For (x = 1), (1^3 - 2(1) > 1), which is false. Therefore, the interval ((0.62, 1.70)) is not part of the solution.

For (x = 2), (2^3 - 2(2) > 1), which is true. Therefore, the interval ((1.70, ∞)) is part of the solution.

Thus, the solution to the inequality (x^3 - 2x > 1) is ((-∞, -1.32) ∪ (1.70, ∞)).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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