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Quadratic Inequalities

Author: Sophia

what's covered
In this lesson, you will learn how to determine the solution set to a quadratic inequality. Specifically, this lesson will cover:

Table of Contents

1. Review of 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.


2. Solutions to Quadratic Inequalities

When we find solutions to a quadratic inequality, we are looking for all x-values that make the inequality statement true. Let's take a look at an example.

EXAMPLE

Consider this quadratic inequality 2 x squared plus 3 x minus 7 less than 4.

Solutions to this inequality are all x-values that make this inequality statement true. That is, there is a set of x-values that makes 2 x squared plus 3 x minus 7 less than 4, and there is a set of x-values that makes 2 x squared plus 3 x minus 7 greater than or equal to 4. 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.


3. Solving a Quadratic Inequality

We generally follow these steps to solve a quadratic inequality.

step by step
  1. Rewrite as an equation set equal to zero.
  2. Solve the equation using which ever method you prefer (factoring, quadratic formula, or completing the square).
  3. Use solutions to create intervals on a number line.
  4. Choose a test value that falls within each interval on the number line.
  5. 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 define 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.

EXAMPLE

Find the solutions for the quadratic inequality x squared plus 5 x – 6 greater than space 8.

First, we need to write this as 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:

x squared plus 5 x minus 6 greater than 8 Rewrite as an equation set equal to zero
x squared plus 5 x minus 6 equals 8 Subtract 8 from both sides
x squared plus 5 x minus 14 equals 0 Factor the equation
open parentheses x minus 2 close parentheses open parentheses x plus 7 close parentheses equals 0 Set each factor equal to zero
x minus 2 equals 0 comma space space space x plus 7 equals 0 Evaluate
x equals 2 comma space space space x equals short dash 7 Solutions to the equation

Now, use these solutions to 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, like -8, 0, or 4. If possible, avoid using decimals. We are going to use them as x-values to be plugged into the inequality.

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. We can rewrite the original inequality x squared plus 5 x minus 6 greater than 8 as x squared plus 5 x minus 14 greater than 0.

Using this rewritten inequality, choose test values within the three intervals. We need a test value that is less than -7, a test value that is between -7 and 2, and a test value greater than 2. Let's use -8, 0, and 4 and plug them into the inequality



x bold italic x to the power of bold 2 bold plus bold 5 bold italic x bold minus bold 14 bold greater than bold 0 Result Interval
-8 table attributes columnalign left end attributes row cell open parentheses short dash 8 close parentheses squared plus 5 open parentheses short dash 8 close parentheses minus 14 greater than 0 end cell row cell 64 minus 40 minus 14 greater than 0 end cell row cell 10 greater than 0 end cell end table This inequality is TRUE, so this interval is part of the solution. x less than short dash 7
0 table attributes columnalign left end attributes row cell 0 squared plus 5 open parentheses 0 close parentheses minus 14 greater than 0 end cell row cell 0 plus 0 minus 14 greater than 0 end cell row cell short dash 14 greater than 0 end cell end table This inequality is NOT TRUE, so this interval is NOT part of the solution. short dash 7 less or equal than x less or equal than 2
4 table attributes columnalign left end attributes row cell 4 squared plus 5 open parentheses 4 close parentheses minus 14 greater than 0 end cell row cell 16 plus 20 minus 14 greater than 0 end cell row cell 22 greater than 0 end cell end table This inequality is TRUE, so this interval is part of the solution. x greater than 2

The values x less than short dash 7 and x greater than 2 are included in the solution region, while the values short dash 7 less or equal than x less or equal than 2 make up the non-solution region. Note that we used less than or greater than symbols with our solutions. This is because the original inequality x squared plus 5 x – 6 greater than space 8 was a strict inequality, meaning that these points are not part of the inequality. We can write the solution to the inequality as:

x less than short dash 7 space space OR space space x greater than 2

summary
Recall that inequalities are statements that two quantities are not equal to each other. The solution to quadratic inequalities is a range of x values that make the inequality statement true. The process for solving a quadratic inequality is 1) solve as an equation set equal to 0; 2) use solutions to the equation to create intervals on a number line; 3) choose a test value that falls within each interval on the number line; and 4) plug each test value into the inequality to identify the solution regions.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License