Solve System of Inequalities $ y (x + 1)^2 - 4 $ and $ y

Question

Sketch the solution to the system of inequalities $ y > (x + 1)^2 - 4 $ and $ y < 2x + 1 $, then identify one solution to the system.

Accepted Answer

One solution is (1, 2) (other valid points exist).

Answer

1. Analyze $ y > (x + 1)^2 - 4 $: The quadratic $ (x + 1)^2 - 4 $ is a parabola opening upward with vertex at $(-1, -4)$. The inequality $ > $ means the parabola is dashed, and we shade above it. 2. Analyze $ y < 2x + 1 $: This is a linear line with slope $ 2 $ and $ y $-intercept $ 1 $. The inequality $ < $ means the line is dashed, and we shade below it. 3. Identify the overlapping region (solution): The solution is where both shadings overlap. Test the point $(1, 2)$: - For $ y > (1 + 1)^2 - 4 $: $ 2 > 4 - 4 $ → $ 2 > 0 $ (true). - For $ y < 2(1) + 1 $: $ 2 < 3 $ (true). Thus, $(1, 2)$ is a solution.