1.
(e^x - 1 )/(x^2 - 1) < 0
2.
The inverse of ln (3 - x) ?
3.
Simplify x^3 - 3x^2 + 2 = 0
Any help would be GREATLY appreciated. Thank you so much for your time.
1. To solve the inequality (e^x - 1)/(x^2 - 1) < 0, we can start by finding the critical points where the numerator or denominator becomes zero.
For the numerator, e^x - 1 = 0, we can add 1 to both sides to get e^x = 1. Taking the natural logarithm of both sides, we find x = ln(1) = 0.
For the denominator, x^2 - 1 = 0, we can factor to get (x - 1)(x + 1) = 0. So, x = 1 or x = -1.
Next, we need to check the sign of the expression (e^x - 1)/(x^2 - 1) in different intervals between the critical points.
For x < -1, choose a value, say x = -2. Plug it into the expression, (e^-2 - 1)/((-2)^2 - 1) = (e^-2 - 1)/(3), which is positive.
For -1 < x < 0, choose a value, say x = -0.5. Plug it into the expression, (e^-0.5 - 1)/((-0.5)^2 - 1) = (e^-0.5 - 1)/(0.75), which is negative.
For 0 < x < 1, choose a value, say x = 0.5. Plug it into the expression, (e^0.5 - 1)/((0.5)^2 - 1) = (e^0.5 - 1)/(-0.75), which is positive.
For x > 1, choose a value, say x = 2. Plug it into the expression, (e^2 - 1)/(2^2 - 1) = (e^2 - 1)/(3), which is positive.
So, the inequality (e^x - 1)/(x^2 - 1) < 0 is satisfied for -1 < x < 0.
2. To find the inverse of ln(3 - x), we can start by replacing ln(3 - x) with y and then switch the roles of x and y.
So, y = ln(3 - x)
Next, solve for x in terms of y.
e^y = 3 - x
Rearrange the equation to solve for x.
x = 3 - e^y
So, the inverse of ln(3 - x) is x = 3 - e^y.
3. To simplify the equation x^3 - 3x^2 + 2 = 0, we can start by looking for any possible rational roots using the rational root theorem.
Possible rational roots are of the form ±(factor of constant term) / (factor of leading coefficient).
The constant term is 2, which has factors ±1 and ±2. The leading coefficient is 1, which only has factors ±1.
Possible rational roots are ±1 and ±2.
We can test these possible roots by substituting each one into the equation to check if it results in an equality.
If we try x = 1, we get 1^3 - 3(1^2) + 2 = 0, which is true.
Therefore, one root is x = 1.
Next, we divide the polynomial x^3 - 3x^2 + 2 by the binomial (x - 1) using long division or synthetic division to find the remaining quadratic factor.
Dividing x^3 - 3x^2 + 2 by (x - 1), we get x^2 - 2x - 2.
Now we have the quadratic factor x^2 - 2x - 2 = 0, which can be solved using the quadratic formula or factoring.
Using the quadratic formula, x = (-(-2) ± √((-2)^2 - 4(1)(-2))) / (2(1)).
Simplifying further, x = (2 ± √(4 + 8)) / 2.
x = (2 ± √12) / 2.
x = (2 ± 2√3) / 2.
x = 1 ± √3.
So, the solutions to the equation x^3 - 3x^2 + 2 = 0 are x = 1, x = 1 + √3, and x = 1 - √3.