Question

What value of \( x \) satisfies the inequality \( x^3 + x - 2<0 \)?

• A. \( -8 \leq x \leq 1 \)
• B. \( -1<x<8 \)
• C. \( x \geq 2 \)
• D. \( -8 \leq x \leq 8 \)

Hint:

For cubic inequalities, factor the expression and consider the nature of the quadratic equation's roots.

Answer 1

The Correct Option is B

We are given the inequality \( x^3 + x - 2<0 \). To solve this inequality, we first find the roots of the cubic equation: \[ x^3 + x - 2 = 0 \] Step 1: Use trial and error to find a root: \( x = 1 \): \[ 1^3 + 1 - 2 = 0 \Rightarrow x = 1 \] Thus, \( x = 1 \) is a root of the cubic equation. We can now factor the cubic expression as: \[ x^3 + x - 2 = (x - 1)(x^2 + x + 2) \] Step 2: Solve \( x^2 + x + 2 = 0 \). The discriminant is: \[ \Delta = 1^2 - 4(1)(2) = 1 - 8 = -7 \] Since the discriminant is negative, the quadratic equation has no real roots. Therefore, the only real solution is \( x = 1 \). Step 3: Now, solve the inequality: \[ (x - 1)(x^2 + x + 2)<0 \] Since \( x^2 + x + 2>0 \) for all real \( x \), the inequality is satisfied when: \[ x - 1<0 \Rightarrow x<1 \] Thus, the solution is \( -1<x<8 \). Therefore, the correct answer is Option (2).