Big Ideas Math Algebra 2, 2014
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Big Ideas Math Algebra 2, 2014 View details
6. Quadratic Inequalities
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Exercise 2 Page 144

For the graphic method, start by graphing the associated quadratic function. For the analytic method, start by replacing the inequality symbol with an equal sign, then find the roots for the equation obtained.

See solution.

Practice makes perfect
We can write a quadratic inequality in one of the following forms, where and are real numbers and
One of the ways to solve a quadratic inequality is by using a graph of the associated function. This is the graphical method. Another way is by using the algebraic method. For this, we need to solve the associated quadratic equation and some test values. We will illustrate how to solve the inequality shown below using both methods.

Solving the Quadratic Inequality by Graphing

Let's start by graphing the corresponding quadratic function. This will allow us to identify the values for which the quadratic inequality holds true.

For this inequality we are just interested in the for which the associated function takes negative values. Let's identify them in the graph.

As we can see, the function goes below the approximately at Then, it starts going above it at approximately Therefore, we can approximate the solution of the inequality set as

Solving the Quadratic Inequality by the Algebraic Method

First, we need to write and solve the equation obtained by replacing with
We can solve this equation by using the Quadratic Formula.
Let's identify the parameters.
Now we can proceed by substituting them into the Quadratic Formula to find the solutions.
Simplify right-hand side
We can find an approximated form for the solutions using the expression found above.

Now we know the critical values of the inequality. We can plot them on a number line using open points. This is because these values do not satisfy the inequality, since the inequality symbol is Note that the critical values partition the number line into three intervals.

We can use a test value from each of these intervals to determine whether they satisfy the inequality or not.

Interval Interval Interval
Test Value
Substitute in
Is it a solution?

Now that we know which intervals belong to the solution set and which of them do not, we can graph the inequality.