Solve System of Inequalities $ y 2x^2 - x - 1 $ and $ y -3x + 2 $ (Algebra 2)

Question

Sketch the solution to the system of inequalities $ y > 2x^2 - x - 1 $ and $ y > -3x + 2 $, then identify one solution to the system.

Accepted Answer

One solution is (0, 3) (other valid points exist).

Answer

1. Analyze $ y > 2x^2 - x - 1 $: The quadratic $ 2x^2 - x - 1 $ has vertex at $ \left( \frac{1}{4}, -1.125 \right) $ (opening upward). The inequality $ > $ means the parabola is dashed, and we shade above it. 2. Analyze $ y > -3x + 2 $: This is a linear line with slope $ -3 $ and $ y $-intercept $ 2 $. The inequality $ > $ means the line is dashed, and we shade above it. 3. Identify the overlapping region (solution): The solution is where both shadings (above both the parabola and the line) overlap. Test the point $(0, 3)$:    - For $ y > 2(0)^2 - 0 - 1 $: $ 3 > -1 $ (true).    - For $ y > -3(0) + 2 $: $ 3 > 2 $ (true). Thus, $(0, 3)$ is a solution.