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Quadratic Inequalities

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- Inequalities
- Solutions to Quadratic Inequalities
- Solving a Quadratic Inequality

**Inequalities**

An inequality is a statement that two quantities are not equal to each other. In general, we see statements that use inequality symbols to show that one quantity is greater than or less than another. However, inequality symbols can be strict or non-strict. The distinction here is that non-strict inequalities "allow" the two quantities to be exactly equal to each other, while strict inequality symbols do not. This is the difference between "greater than" (>) and "greater than or equal to" (≥), for example. It is also important to remember that when graphing inequalities and when plotting solutions on a number line, we use open circles, curved braces, and dotted lines for strict inequalities; and we use closed circles, square brackets, and solid lines for non-strict inequalities.

**Solutions to Quadratic Inequalities**

Let's consider this quadratic inequality: 2x^{2} + 3x – 7 < 4

Solutions to this inequality are all x-values that makes this inequality statement true. That is, there is a set of x-values that makes 2x^{2} + 3x – 7 less than 4, and there is a set of x-values that makes 2x^{2} + 3x – 7 greater than or equal to 7. The former is the solution set to the inequality, since it makes our statement true, while the latter is the set of all non-solutions, because it makes the inequality statement false.

To find solutions to a quadratic inequality, it is often helpful to first think of the relationship as an equation, and then consider the inequality once solutions to the equation have been found. This is because we have a variety of tools at our disposal to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula.

**Solving a Quadratic Inequality**

We generally follow these steps to solve a quadratic inequality.

- Rewrite as an equation set equal to zero
- Solve the equation (using factoring, completing the square, or quadratic formula - whichever method you prefer)
- Use solutions to create intervals on a number line
- Choose a test value that falls within each interval on the number line
- Plug each test value into the inequality (with zero on one side) to identify solution regions

In other words, when we treat the inequality as an equation and find solutions to the equation, we identify critical points to defining the solution region. We then choose any value we want within certain intervals (defined by these critical points) and see if they yield true or false statements to the original inequality. If a test point satisfies the inequality, the interval it lies within is included in our solution region. If a test point does not satisfy the inequality, the interval it lies within is excluded from the solution region.

Solve the quadratic inequality x^{2} + 5x – 6 > 8

First, we need to write this an equation, and set it equal to zero. This means we will use an equals sign instead of an inequality symbol, and then subtract 8 from both sides. Then we can solve the quadratic equation:

These solutions create three intervals on the number line:

Next, we choose any value that fits within each interval. It doesn't matter which values we choose to be test values, but make it as simple as possible. We are going to use them as x-values to be plugged into the inequality (with zero on one side)

We can certainly plug these values into the original inequality that has 8 on one side of the inequality symbol. However, it makes the process easier to compare values to zero. This is because we just need to determine if the value is positive or negative to decide if it satisfies the inequality or not.

Let's choose the test values: –8, 0, and 3.

Since our inequality symbol is "greater than" we looked to see which test points yield positive y–values. The intervals and are included in the solution region, while the interval makes up the non-solution region.

We can write the solution to the inequality as: