Quadratic Inequalities Tidewater Community College Karen Overman

Quadratics Before we get started let’s review. A quadratic equation is an equation that can be written in the form , where a, b and c are real numbers and a cannot equal zero. In this lesson we are going to discuss quadratic inequalities.

Quadratic Inequalities What do they look like? Here are some examples:

Quadratic Inequalities When solving inequalities we are trying to find all possible values of the variable which will make the inequality true. Consider the inequality We are trying to find all the values of x for which the quadratic is greater than zero or positive.

Solving a quadratic inequality We can find the values where the quadratic equals zero by solving the equation,

Solving a quadratic inequality You may recall the graph of a quadratic function is a parabola and the values we just found are the zeros or x-intercepts. The graph of is

Solving a quadratic inequality From the graph we can see that in the intervals around the zeros, the graph is either above the x-axis (positive) or below the x-axis (negative). So we can see from the graph the interval or intervals where the inequality is positive. But how can we find this out without graphing the quadratic? We can simply test the intervals around the zeros in the quadratic inequality and determine which make the inequality true.

Solving a quadratic inequality For the quadratic inequality, we found zeros 3 and –2 by solving the equation . Put these values on a number line and we can see three intervals that we will test in the inequality. We will test one value from each interval. -2 3

Solving a quadratic inequality Interval Test Point Evaluate in the inequality True/False

Solving a quadratic inequality Thus the intervals make up the solution set for the quadratic inequality, . In summary, one way to solve quadratic inequalities is to find the zeros and test a value from each of the intervals surrounding the zeros to determine which intervals make the inequality true.

Example 2: Solve First find the zeros by solving the equation,

Example 2: Now consider the intervals around the zeros and test a value from each interval in the inequality. The intervals can be seen by putting the zeros on a number line. 1/2 1

Example 2: Interval Test Point Evaluate in Inequality True/False

Example 2: Thus the interval makes up the solution set for the inequality .

Example 3: Solve the inequality . First find the zeros.

Example 3: But these zeros , are complex numbers. What does this mean? Let’s look at the graph of the quadratic,

Example 3: We can see from the graph of the quadratic that the curve never intersects the x-axis and the parabola is entirely below the x-axis. Thus the inequality is always true.

Example 3: How would you get the answer without the graph? The complex zeros tell us that there are no REAL zeros, so the parabola is entirely above or below the x-axis. At this point you can test any number in the inequality, If it is true, then the inequality is always true. If it is false, then the inequality is always false. We can also determine whether the parabola opens up or down by the leading coefficient and this will tell us if the parabola is above or below the x-axis.

Summary In general, when solving quadratic inequalities Find the zeros by solving the equation you get when you replace the inequality symbol with an equals. Find the intervals around the zeros using a number line and test a value from each interval in the number line. The solution is the interval or intervals which make the inequality true.

Practice Problems