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The quadratic formula is a key tool used to solve quadratic equations, such as the equation from our exercise, which is of the form ax^2 + bx + c = 0. This formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), provides the values of x that make the equation true by setting the left side equal to zero. Quadratic equations are remarkable in that they always have two solutions, which can be real or complex numbers.

When working with inequalities, these solutions act as critical points that divide the number line into intervals. Each interval represents a range of x values that we must test to find where the inequality is satisfied. It's important to use accurate calculations when applying the quadratic formula since these solutions are crucial for the next steps in solving inequalities. An error here could lead to an incorrect assessment of the inequality's solution set.

Graphing on the real number line is a visual approach to understanding solutions to inequalities. Once we have solved the quadratic equation using the formula, we plot the solutions as points on the number line. For the inequality \(2 x^{2}+3 x-35<0\), the points we get from solving \(2 x^{2}+3 x-35 = 0\) mark where the sign of the expression changes.

These points divide the number line into distinct intervals. A filled-in circle is used to represent a solution that includes the endpoint, whereas an open circle is used when the endpoint is not included. In our example, open circles are used because the inequality is 'less than'. The graph provides a clear and immediate visualisation of the ranges for x that satisfy the original inequality, and it is extremely helpful for confirming the results of our interval tests.

Interval testing is the process of checking each interval identified by the solutions on the number line to determine where the initial inequality holds true. After calculating the solutions of the quadratic equation and placing them on the number line, we select a test point from each interval that does not include the solutions themselves.

By substituting these test points into the original inequality, we can determine whether the inequality is satisfied in that interval. If it is, the entire interval is part of the solution set. The process involves trial and error, but with the visual aid of the number line graph and strategically chosen test points, we reduce the number of steps and streamline the solution process. Clear understanding and careful calculations ensure the accuracy of the interval testing, ultimately leading us to correctly determine the solution set of the original inequality.