Question

Give an example of each of the types of quadratic inequalities.

Short answer

Examples of quadratic inequalities are \(x^2 - 5x + 6 < 0\) and \(x^2 - 7x + 10 > 0\). The '<' inequality is true for all \(x\) in the range \(2 < x < 3\) and the '>' inequality is true for all \(x < 2\) or \(x > 5\)

Step-by-step solution

Set up '

Let's create a quadratic inequality in the form of '<' or '<='. We will use the quadratic function \(x^2 - 5x + 6 < 0\). This is a valid quadratic inequality as it's a quadratic expression set to be less than zero.

Solve '

The above inequality can be solved by factoring the quadratic expression and determining the values for which the inequality holds true. In this case, the factored form of the quadratic inequality is \((x-3)(x-2) < 0\). By solving this inequality, it can be determined that the solution is \(2 < x < 3\). So for values of x between 2 and 3, the inequality holds true.

Set up '>' or '>=' form of quadratic Inequality

Next, let's create a quadratic inequality in the form of '>' or '>=. For this, we will use the quadratic function \(x^2 - 7x + 10 > 0\). This inequality involves a quadratic expression set to be greater than zero and is a valid quadratic inequality.

Solve '>' or '>=' form of quadratic Inequality

The above inequality can be solved by factoring the quadratic expression and determining the values for which the inequality holds true. In this case, the factored form of the quadratic inequality is \((x-5)(x-2) > 0\). By solving this inequality, it can be determined that the solutions are \(x < 2\) or \(x > 5\). So for values of x that are less than 2 or greater than 5, the inequality holds true.