Questions: Solve the following inequality. x^2 + x ≥ 2 Select the correct choice below and, if necessary, fill in the answer box. A. The solution set is (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) B. There is no real solution

Solution Steps

To solve the inequality \(x^2 + x \geq 2\), we can follow these steps:

1. Rewrite the inequality in standard form: \(x^2 + x - 2 \geq 0\).
2. Factor the quadratic expression: \((x + 2)(x - 1) \geq 0\).
3. Determine the critical points by setting each factor to zero: \(x = -2\) and \(x = 1\).
4. Test the intervals determined by the critical points to see where the inequality holds true.
5. Combine the intervals where the inequality is satisfied and express the solution in interval notation.

We start with the inequality: \[ x^2 + x \geq 2 \] Rearranging gives us: \[ x^2 + x - 2 \geq 0 \]

Next, we factor the quadratic: \[ (x + 2)(x - 1) \geq 0 \]

Setting each factor to zero, we find the critical points: \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

The critical points divide the number line into intervals: \((-\infty, -2)\), \((-2, 1)\), and \((1, \infty)\). We test each interval to determine where the inequality holds.

• For \(x < -2\) (e.g., \(x = -3\)): \((x + 2)(x - 1) = (-)(-) > 0\)
• For \(-2 < x < 1\) (e.g., \(x = 0\)): \((x + 2)(x - 1) = (+)(-) < 0\)
• For \(x > 1\) (e.g., \(x = 2\)): \((x + 2)(x - 1) = (+)(+) > 0\)

The inequality holds true in the intervals: \[ (-\infty, -2) \cup (1, \infty) \]

Final Answer