Questions: What's the solution to the inequality x^2 + 5x ≥ 6?

Transcript text: What's the solution to the inequality $x^{2}+5 x \geq 6$ ?

Solution

Solution Steps

Step 1: Factor the Polynomial

The given inequality is $ x^2 + 5x \geq 6 $. We first rearrange it to standard form:

\[ x^2 + 5x - 6 \geq 0 \]

Next, we factor the polynomial $ x^2 + 5x - 6 $:

\[ x^2 + 5x - 6 = (x - 1)(x + 6) \]

Step 2: Solve the Inequality

We now solve the inequality:

\[ (x - 1)(x + 6) \geq 0 \]

To find the solution, we determine the critical points by setting each factor to zero:

$ x - 1 = 0 $ gives $ x = 1 $
$ x + 6 = 0 $ gives $ x = -6 $

Next, we analyze the sign of the product $ (x - 1)(x + 6) $ in the intervals determined by these critical points: $ (-\infty, -6) $, $ (-6, 1) $, and $ (1, \infty) $.

For $ x < -6 $: Both factors are negative, so the product is positive.
For $ -6 < x < 1 $: One factor is negative and the other is positive, so the product is negative.
For $ x > 1 $: Both factors are positive, so the product is positive.

Thus, the solution to the inequality is:

\[ x \in (-\infty, -6] \cup [1, \infty) \]