What is the Solution Set of the Quadratic Inequality

In algebra, solving a quadratic inequality is very like to solving a quadratic equation. The departure is that with quadratic equations, yous set the expressions equal to aught, but with inequalities, you’re interested in what’s on either side of the naught (positives and negatives).

To solve a quadratic inequality, you lot follow these steps:

1. Movement all the terms to one side of the inequality sign.

2. Cistron, if possible.

3. Decide all zeros (roots, or solutions).

   Zeros are the values of the variable that make each factored expression equal to zero.

4. Put the zeros in order on a number line.

5. Create a sign line to show where the expression in the inequality is positive or negative.

   A sign line shows the signs of the different factors in each interval. If the expression is factored, show the signs of the private factors.

6. Make up one’s mind the solution, writing information technology in inequality notation or interval notation.

Hither’due south an example: Solve the inequality
x
²
–
10
> 12.

Here, y’all need to determine what values of
x
y’all can square so that when you subtract the original number, your answer will exist bigger than 12.

Follow the steps:

1. Subtract 12 from each side of the inequality
   ten
   ²
   –
   x
   > 12 to move all the terms to one side.

   You lot cease up with
   10
   ²
   –
   x
   – 12 > 0.

2. Factoring on the left side of the inequality, you lot get (10
   – 4)(x
   + 3) > 0.

3. Determine that all the zeroes for the inequality are
   x
   = 4 and
   x
   = –3.

4. Put the zeros in order on a number line, shown in the following effigy.

   

5. Create a sign line to show the signs of the different factors in each interval.

   Between –iii and 4, try letting
   ten
   = 0 (you tin utilize any number between –3 and iv). When
   x
   = 0, the factor (ten
   – 4) is negative, and the gene (x
   + 3) is positive. Put those signs on the sign line to represent to the factors. Practice the same for the interval of numbers to the left of –3 and to the correct of iv (run across the side by side analogy).

   

   Go along in heed that the
   10
   values in each interval are really random choices (as you lot can see from the choices in this example of
   x
   = –5 and
   x
   = 10). Whatsoever number in each of the intervals gives y’all the same positive or negative value to the gene.

6. To determine the solution, await at the signs of the factors; you want the expression to exist positive, corresponding to the inequality greater than zero.

   The interval to the left of –3 has a negative times a negative, which is positive. So, any number to the left of –iii works. You tin can write that office of the solution as
   10
   < –3 or, in interval notation,

   

   The interval to the correct of 4 has a positive times a positive, which is positive. So,
   10
   > four is a solution; you can write it as

   

   The interval between –3 and 4 is always negative; yous have a negative times a positive. The complete solution lists both intervals that have working values in the inequality.

   The solution of the inequality
   x
   ^two
   –
   x
   > 12, therefore, is
   ten
   < –3 or
   x
   > 4. Writing this result in interval notation, y’all replace the word “or” with the symbol

   

   and write it as