# Algebra II : Quadratic Inequalities

## Example Questions

### Example Question #1 : Quadratic Inequalities

Explanation:

The problem is already in standard form, so all we have to at first do is set the quadratic expression = 0 and factor as normal.

Negative x^2's are hard to work with, so we multiply through by -1.

Now we can factor easily.

By the zero product property, each of these factors will be equal to 0.

Since -9 and 1 are our zeros, we just have to test one point in the region between them to find out which region our answer set goes in. Let's test x = 0 in the original inequality.

Since this statement is false, the region between -9 and 1 is not correct. So it must be the region on either side of those points. Since the original inequality was less than or equal to, the boundary points are included. So all values from -infinity to -9 inclusive, and from 1 inclusive to infinity, are solutions. In interval notation we write this as:

### Example Question #197 : Solving Quadratic Equations

Explanation:

First we want to rewrite the quadratic in standard form:

Now we want to set it = 0 and factor and solve like normal.

Using the zero product property, both factors produce a zero:

So the two zeros are -2 and 3, and will mark the boundaries of our answer interval. To find out if the interval is between -2 and 3, or on either side, we simply take a test point between -2 and 3 (for instance, x = 0) and evaluate the original inequality.

Since the above is a true statement, we know that the solution interval is between -2 and 3, the same region where we picked our test point. Since the original inequality was less than or equal, we include the endpoints.

Ergo, .

### Example Question #2 : Quadratic Inequalities

What is the discriminant of the following quadratic equation:

Explanation:

The discriminant of a quadratic equation in  form is equal to . The given equation is not in that form however, so we must first multiply it out to get it into that form. We therefore obtain:

We therefore have , and . Our discriminant is therefore:

### Example Question #3 : Quadratic Inequalities

and

Explanation:

1. Rewrite the equation in standard form.

2. Set the equation equal to  and solve by factoring.

So,  and  are our zeroes.

3. Test a point between your zeroes to find out if the solution interval is between them or on either side of them. (Try testing  by plugging it into your original inequality.)

Because the above statement is true, the solution is the interval between  and .

### Example Question #4 : Quadratic Inequalities

Solve this inequality.

Explanation:

Combine like terms first.

Factor

The zeroes are 3 and 8 so a number line can be divided into 3 sections.

X<3 works, 3<x<8 does not work, and x>8 works

### Example Question #5 : Quadratic Inequalities

Solve:

Explanation:

Start by setting the inequality to zero and by solving for .

Now, plot these two points on to a number line.

Notice that these two numbers effectively divide up the number line into three regions:

, and

Now, choose a number in each of these regions and put it back in the factored inequality to see which cases are true.

For , let

Since this  is not less than , the solution to this inequality cannot lie in this region.

For , let .

Since this will make the inequality true, the solution can lie in this region.

Finally, for , let

Since this number is not less than zero, the solution cannot lie in this region.

Thus, the solution to this inequality is

### Example Question #6 : Quadratic Inequalities

Solve:

The solution cannot be determined with the information given.

Explanation:

First, set the inequality to zero and solve for .

Now, plot these two numbers on to a number line.

Notice how these numbers divide the number line into three regions:

Now, you will choose a number from each of these regions to test to plug back into the inequality to see if the inequality holds true.

For , let

Since this is not less than zero, the solution to the inequality cannot be found in this region.

For , let

Since this is less than zero, the solution is found in this region.

For , let

Since this is not less than zero, the solution is not found in this region.

Then, the solution for this inequality is

### Example Question #7 : Quadratic Inequalities

Solve:

Explanation:

Start by changing the less than sign to an equal sign and solve for .

Now, plot these two numbers on a number line.

Notice how the number line is divided into three regions:

Now, choose a number fromeach of these regions to plug back into the inequality to test if the inequality holds.

For , let

Since this number is not less than zero, the solution cannot be found in this region.

For , let

Since this number is less than zero, the solution can be found in this region.

For  let .

Since this number is not less than zero, the solution cannot be found in this region.

Because the solution is only negative in the interval , that must be the solution.

### Example Question #8 : Quadratic Inequalities

Solve:

Explanation:

First, set the inequality to zero and solve for .

Now, plot these two numbers on to a number line.

Notice how these numbers divide the number line into three regions:

Now, you will choose a number from each of these regions to test to plug back into the inequality to see if the inequality holds true.

For , let

Since this solution is greater than or equal to , the solution can be found in this region.

For , let

Since this is less than or equal to , the solution cannot be found in this region.

For , let

Since this is greater than or equal to , the solution can be found in this region.

Because the solution can be found in every single region, the answer to this inequality is

### Example Question #9 : Quadratic Inequalities

Which value for  would satisfy the inequality ?

Not enough information to solve

Explanation:

First, we can factor the quadratic to give us a better understanding of its graph. Factoring gives us: . Now we know that the quadratic has zeros at  and . Furthermore this information reveals that the quadratic is positive. Using this information, we can sketch a graph like this:

We can see that the parabola is below the x-axis (in other words, less than ) between these two zeros  and .

The only x-value satisfying the inequality  is .

The value of  would work if the inequality were inclusive, but since it is strictly less than instead of less than or equal to , that value will not work.