Final answer:
The inequality x² + 4x + 3 ≥ 0 is solved algebraically by factoring to find that the solutions to the inequality are x ≥ -3.
Step-by-step explanation:
The inequality given is x² + 4x + 3 ≥ 0. To solve it algebraically, we can factor the quadratic equation or use the quadratic formula. The equation factors to (x + 1)(x + 3) ≥ 0. This shows us that the solutions where the inequality equals zero are x = -1 and x = -3. Since it's a ≥ inequality, we also include the regions where the parabola is above the x-axis.
The solutions to the inequality are obtained by evaluating the signs of the factors within different intervals. These intervals in this case are x < -3, -3 ≤ x ≤ -1, and x > -1. Through testing points in each interval, we determine that the inequality holds true for x values in the intervals [-3, -1] and x > -1. Thus, the final solution is x ≥ -3.
We did not need to use the quadratic formula here, but for reference, for any quadratic equation of the form ax² + bx + c = 0, the solutions can be found using the quadratic formula x = (-b ± √(b² - 4ac)) / (2a).