Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the inequality. $$y^{2}-x<0$$

Step-by-Step Solution

Answer

The graph of the inequality \(y^2 - x < 0\) is the region within the 'V' shape formed by the curves \(y = \sqrt{x}\) and \(y = - \sqrt{x}\). The region does not include the boundary.

Step 1: Sketch the Boundary

The boundary of the region is given by the equality \(y^2 - x = 0\). Rearrange this equation to \(y = \pm \sqrt{x}\). Now, sketch these two functions \(y = \sqrt{x}\) and \(y = - \sqrt{x}\) on a coordinate plane. This forms a V shape.

Step 2: Test a point

Choose a point not on the boundary to test which side of the boundary to shade. A good choice is the origin (0,0). Substituting these values into the inequality \(y^2 - x < 0\) gives \(0 - 0 < 0\) which is false. Therefore, do not shade the region that contains the origin.

Step 3: Shade the Correct Region

Since the region containing the origin is not shaded, the region that is shaded is the one that does not contain the origin, i.e., the region within the 'V' shape formed by the two curves \(y = \sqrt{x}\) and \(y = - \sqrt{x}\).