Solve System of Inequalities $ y \leq x^2 + 2x + 1 $ and $ y x - 1 $ (Algebra 2)

Question

Sketch the solution to the system of inequalities $ y \leq x^2 + 2x + 1 $ and $ y > x - 1 $, then identify one solution to the system.

Accepted Answer

One solution is (0, 0) (other valid points exist).

Answer

1. Analyze $ y \leq x^2 + 2x + 1 $: The quadratic $ x^2 + 2x + 1 = (x + 1)^2 $ is a parabola opening upward with vertex at $(-1, 0)$. The inequality $ \leq $ means the parabola is solid, and we shade below it. 2. Analyze $ y > x - 1 $: This is a linear line with slope $ 1 $ and $ y $-intercept $ -1 $. The inequality $ > $ means the line is dashed, and we shade above it. 3. Identify the overlapping region (solution): The solution is where both shadings overlap. Test the point $(0, 0)$:    - For $ y \leq (0)^2 + 2(0) + 1 $: $ 0 \leq 1 $ (true).    - For $ y > 0 - 1 $: $ 0 > -1 $ (true). Thus, $(0, 0)$ is a solution.