# What is the Solution Set of the Quadratic Inequality Mc006-1.jpg

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A quadratic inequality is one that includes an

${\displaystyle x^{2}}$

x

2

{\displaystyle x^{2}}

term and thus has two roots, or two x-intercepts. This results in a parabola when plotting the inequality on a coordinate plane. Solving an inequality means finding the values of x that make the inequality true. You can show these solutions algebraically, or by illustrating the inequality on a number line or coordinate plane.

1. 1

2. 2

Find two factors whose product is the first term of the inequality.
To factor the inequality, you need to find two binomials whose product equals the standard form of the inequality. A binomial is a two-termed expression.[2]
To do this, you need to complete the FOIL method in reverse. Begin by finding two factors for the first term of each binomial.

3. 3

Find two factors whose product is the third term in the standard form of the inequality.
These two factors must also have a sum equal to the second term in the inequality. You will likely need to do some guess-and-check work at this time, to see which two factors meet these two requirements. Make sure you pay close attention to the positive and negative signs as well.

• For example:

1. 1

Determine whether your factors have the same sign.
If, according to the inequality, the product of the factors is greater than zero, then either both factors will be negative (less than 0), or both factors will be positive (greater than 0), since a negative times a negative equals a positive, and a positive times a positive equals a positive.[3]

2. 2

Determine whether your factors have opposite signs.
If, according to the inequality, the product of the factors is less than 0, then one factor will be less than 0, or negative, and the other factor will be greater than zero, or positive. This is because a negative times a positive equals a negative.

3. 3

Write out the options for the roots.
Write these options by turning each factor into an inequality, based on whether they will have the same or opposite signs. You should have two options.
[4]

4. 4

Simplify the roots for the first option.
To simplify, isolate the

${\displaystyle x}$

x

{\displaystyle x}

variable for each factor. Don’t forget that if you multiply or divide an inequality by a negative number, you must flip the inequality sign.[5]

5. 5

Check the validity of the roots for your first option.
To do this, see whether you can combine the roots to make a correct inequality. If you can find values that are true for both roots, then the option is valid. If you can’t, the roots in this option are not valid.[6]

6. 6

Simplify the roots of the second option.
Isolate the

${\displaystyle x}$

x

{\displaystyle x}

variable for each factor, remembering to flip the inequality sign if you multiply or divide by a negative number.[7]

7. 7

Check the validity of the roots for your second option.
If you can find values that are true for both roots, then the option is valid. If you can’t, the roots in this option are not valid.[8]

1. 1

Draw a number line.
Make sure you draw it according to any required specifications. If your number line has no specifications, just make sure to include positions for both

${\displaystyle x}$

x

{\displaystyle x}

values your found previously. Include a few values above and below these to make the number line easier to interpret.

2. 2

3. 3

1. 1

Plot the x-intercepts on the coordinate plane.
An x-intercept is a point where the parabola crosses the x-axis. The two roots you found are the x-intercepts.[10]

2. 2

3. 3

Find the vertex of the parabola.
The vertex is the high or low point of the parabola. To find the vertex, first change the original inequality into an equation equal to

${\displaystyle y}$

y

{\displaystyle y}

. Then plug the

${\displaystyle x}$

x

{\displaystyle x}

value you found for the axis of symmetry into the equation.[12]

4. 4

5. 5

6. 6

To know whether to shade above or below the x-axis, you need to look at the original inequality. If the inequality is less than zero, you will shade below the x-axis. If the inequality is greater than zero, you will shade above the x-axis.[15]
To know whether to shade inside the parabola or outside of the parabola, look at your roots, or your number line. If the valid values of

${\displaystyle x}$

x

{\displaystyle x}

lie between the two roots, you will shade inside the parabola. If the valid values of

${\displaystyle x}$

x

{\displaystyle x}

lie outside the two roots, you will shade outside the parabola.[16]

• For example, since the inequality is

${\displaystyle x^{2}+4x-21<0}$

x

2

+
4
x

21
<

{\displaystyle x^{2}+4x-21<0}

, you will shade a region below the x-axis. Since the valid values lie between the roots -7 and 3, you will shade the region between these two points.

• Question

When do we need to change the greater than and less than symbol of an inequality?

When you divide or multiply an inequality by a negative number, you need to flip the inequality sign.

• Question

How do I solve x in (x2 + 1)2 + 2(x2 + 1) – 35 = 0?

Here’s the easiest way to solve for x: Let a = (x² + 1). Then a² + 2a – 35 = 0. By factoring, (a + 7)(a – 5) = 0. Solving for a, a = -7 or 5. Then (x² + 1) = -7 or 5. If (x² + 1) = -7, x² = -8, and x = +/-√-8 = +/-2i√2 (both “imaginary” numbers). If (x² +1) = 5, x² = 4, and x = +/- 2, (which are “real” numbers). If you prefer, you may reject the imaginary roots, leaving x = +/- 2. Both of the “real” roots work when plugged back into the original equation.

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Article Summary
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To solve a quadratic inequality, first write it as ax^2 + bx + c is less than 0. Then find 2 factors whose product is its first term and 2 factors whose product is its third term. Be sure the 2 factors whose product is its third term also have a sum that’s equal to its second term. Now determine whether your factors have the same or opposite signs by seeing if the product of the factors is greater or less than 0. Finally, turn each factor into an inequality, simplify, and check the validity of the roots for each option.
If you want to learn how to show the solutions on a number line, keep reading the article!

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