Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and graph the solution on a real number line. \(2 x^{2}+3 x-35<0\)

Step-by-Step Solution

Answer

The solution to the quadratic inequality \(2 x^{2}+3 x-35<0\) will be range of values that can be represented on the real number line once the roots and their corresponding intervals have been evaluated for the inequality.

Step 1: Identify the inequality

The inequality given is \(2 x^{2}+3 x-35<0\). The objective is to find the values of x which satisfy this inequality.

Step 2: Solve the corresponding quadratic equation

To find the intervals, solve the equation \(2 x^{2}+3 x-35 = 0\). Apply the quadratic formula \(x= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). On substituting the values a = 2, b = 3, and c = -35, the roots x1 and x2 can be calculated.

Step 3: Determine the intervals

The roots x1 and x2 split the real number line into three intervals: \(x < x1\), \(x1 < x < x2\), and \(x > x2\). Each of these intervals needs to be checked to determine where the inequality holds true.

Step 4: Test each interval

Choose a test point in each interval, substitute it into the inequality, and see if the inequality holds true. Any intervals where the inequality holds true is part of the solution.

Step 5: Graph the solution

Present the solution intervals on a real number line. Open circles are used at x1 and x2 because the inequality is 'less than' and not 'less than or equal to'.

Key Concepts

Quadratic Formula

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.

Real Number Line Graphing

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

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.