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A quadratic inequality is one that includes an 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.
Steps
-
Write the inequality in the standard form. The standard form of a quadratic is a trinomial that follows the structure , where , , and are known coefficients, and . [1] X Research source
- For example, the inequality
is not in standard form. First, you need to use the distributive property to multiply
and
. Then, you need to subtract 21 from both sides of the inequality:
- For example, the inequality
is not in standard form. First, you need to use the distributive property to multiply
and
. Then, you need to subtract 21 from both sides of the inequality:
-
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] X Research source To do this, you need to complete the FOIL method in reverse. Begin by finding two factors for the first term of each binomial.
- For example, , so you can begin setting up your factors like this: .
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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:
- -21 is the third term in the inequality, so these two factors (7 and -3) might work. Now you need to see whether the sum of these factors equals the second term ( ) of the inequality.
- Since , these two factors meet both requirements. So, your factored inequality is .
- For example:
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-
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] X Research source
- If the inequality is greater than or equal to ( ) or less than or equal to ( ), one or both of the factors may be zero.
- For example, for the inequality , the product of the factors is less than 0, and so the two factors will not have the same sign.
-
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.
- Again, if the inequality is greater than or equal to ( ) or less than or equal to ( ), one or both of the factors may be zero.
- For example, for the inequality , the product of the factors is less than 0, and so the two factors will have different signs.
-
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] X Research source
- For example, you found that the factors of the inequality
must have opposite signs, so your options would be stated thusly:
AND (That is, the first factor will be negative, and the second factor will be positive.)
OR
AND (That is, the first factor will be positive, and the second factor will be negative.)
- For example, you found that the factors of the inequality
must have opposite signs, so your options would be stated thusly:
-
Simplify the roots for the first option. To simplify, isolate the 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] X Research source
- For example, the first option for
was that
AND
.
- First, solve
for
:
- Then solve
for
:
- First, solve
for
:
- So, your simplified roots for the first option are and .
- For example, the first option for
was that
AND
.
-
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] X Research source
- For example, for the first option, and , you need to determine whether there are values that satisfy both requirements. Ask yourself, is there a value that is both less than -7 and greater than 3? Since no number can be both less than -7 and greater than 3, you know that this option is not valid.
-
Simplify the roots of the second option. Isolate the variable for each factor, remembering to flip the inequality sign if you multiply or divide by a negative number. [7] X Research source
- For example, the second option for
was that
AND
.
- First, solve
for
:
- Then solve
for
:
- First, solve
for
:
- So, your simplified roots for the second option are and .
- For example, the second option for
was that
AND
.
-
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] X Research source
- For example, the second option is that and , so you need to find a value for that would satisfy both inequalities. Ask yourself, is there a value that is both greater than -7 and less than 3? Since there are many numbers that are both greater than -7 and less than 3 (0, for example), you know that this option is valid, and so these roots are the solution to the inequality.
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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 values your found previously. Include a few values above and below these to make the number line easier to interpret.
- For example, since the roots for the inequality are and , draw a number line that includes positions for -7 and 3.
-
Plot the values on the number line. Plot the points by drawing a circle over their position on the number line. If the inequality is greater than ( ) or less than ( ), draw an open circle. If the inequality is greater than or equal to ( ) or less than or equal to ( ), fill in the circle on the number line, since the values are included in the set. [9] X Research source
- For example, since the roots your are working with are and , you would draw open circles at the -7 and 3 positions on the number line.
-
Draw arrows or lines indicating the included values. If is greater than the value, draw a line pointing to the right on the number line, since the included values will be larger than . If is less than the value, draw a line pointing to the left on the number line, since the included values will be less than . If the included values are between two numbers, you will draw a line between the two plotted points.
- For example, since you want to show that but also , you need to draw a line between -7 and 3 on the number line.
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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] X Research source
- For example, if the inequality is , then the x-intercepts are and , since these are the roots you found when using the quadratic formula or factoring.
-
Find the axis of symmetry. The axis of symmetry is the line that cuts the parabola in half. To find the axis of symmetry, use the formula , where and correspond to the terms in the original quadratic inequality. [11] X Research source
- For example, for the inequality
, you will first calculate
:
. So, the axis of symmetry is the line
- For example, for the inequality
, you will first calculate
:
-
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 . Then plug the value you found for the axis of symmetry into the equation. [12] X Research source
- For example, if the axis of symmetry is
, plug -2 into the equation and solve:
So, the vertex of the parabola is at the point .
- For example, if the axis of symmetry is
, plug -2 into the equation and solve:
-
Determine the direction of the parabola. To know the direction of the parabola, look at the term of the inequality in standard form. If the term is positive, the parabola will “right-side up,” meaning it opens towards the top. If the term is negative, the parabola will be “upside down,” meaning it opens towards the bottom. [13] X Research source
- Since the term in the inequality is positive, the parabola will be right-side up.
-
Draw the parabola with a solid or dotted line. If the inequality is greater than or equal to ( ) or less than or equal to ( ), draw the parabola with a solid line, since the values on the line are included in the solution set. If the inequality is greater than ( ) or less than ( ), draw the parabola with a dotted line, since the values on the line are not included in the solution set. [14] X Research source
- Since the line is less than zero (not less or equal to), you should draw the parabola with a dotted line.
-
Shade the graph. 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] X Research source 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 lie between the two roots, you will shade inside the parabola. If the valid values of lie outside the two roots, you will shade outside the parabola. [16] X Research source
- For example, since the inequality is , 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.
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Community Q&A
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QuestionWhen do we need to change the greater than and less than symbol of an inequality?Community AnswerWhen you divide or multiply an inequality by a negative number, you need to flip the inequality sign.
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QuestionHow do I solve x in (x2 + 1)2 + 2(x2 + 1) - 35 = 0?DonaganTop AnswererHere'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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QuestionWhat is x if (x+2)/(x+1) is greater than 4?Community Answer(x + 2) / (x + 1) > 4. Multiply both sides by (x + 1): (x + 2) > 4(x + 1). Then (x + 2) > 4x + 4. Subtract x on both sides: 2 > 3x + 4. Now subtract 4 on both sides: -2 > 3x. Then divide both sides by 3: -2/3 > x.
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References
- ↑ http://www.mathsisfun.com/algebra/quadratic-equation.html
- ↑ http://www.mathwarehouse.com/dictionary/B-words/what-is-a-binomial.php
- ↑ https://www.khanacademy.org/math/algebra-home/alg-quadratics/alg-quadratic-inequalities/v/quadratic-inequality-example-2
- ↑ https://www.khanacademy.org/math/algebra-home/alg-quadratics/alg-quadratic-inequalities/v/quadratic-inequality-example-2
- ↑ http://www.purplemath.com/modules/ineqsolv.htm
- ↑ https://www.khanacademy.org/math/algebra-home/alg-quadratics/alg-quadratic-inequalities/v/quadratic-inequality-example-2
- ↑ http://www.purplemath.com/modules/ineqsolv.htm
- ↑ https://www.khanacademy.org/math/algebra-home/alg-quadratics/alg-quadratic-inequalities/v/quadratic-inequality-example-2
- ↑ http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/inequalitiesrev4.shtml
- ↑ http://www.themathpage.com/aprecalc/roots-zeros-polynomial.htm
- ↑ http://www.virtualnerd.com/algebra-2/quadratics/inequalities/graphing-solving-inequalities/graph-inequality
- ↑ http://www.virtualnerd.com/algebra-2/quadratics/inequalities/graphing-solving-inequalities/graph-inequality
- ↑ http://www.dummies.com/test-prep/act/act-trick-for-quadratics-how-to-quickly-find-the-direction-of-a-parabola/
- ↑ http://www.varsitytutors.com/hotmath/hotmath_help/topics/graphing-quadratic-inequalities
- ↑ https://www.khanacademy.org/math/algebra-home/alg-quadratics/alg-quadratic-inequalities/v/quadratic-inequalities-visual-explanation
- ↑ http://www.purplemath.com/modules/ineqquad.htm
About This Article
Article Summary
X
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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