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Math: How to Solve a Quadratic Inequality

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I studied applied mathematics, in which I did both a bachelor's and a master's degree.

An inequality is a mathematical expression in which two functions are compared such that the righthand side is either larger or smaller than the lefthand side of the inequality sign. If we do not allow both sides to be equal, we speak of a strict inequality. This gives us four different types of inequalities:

  • Less than: <
  • Less than or equal to: ≤
  • Larger than: >
  • Larger than or equal to ≥

When Is a Quadratic Inequality?

In this article, we will focus on inequalities with one variable, but there can be multiple variables. However, this would make it very difficult to solve by hand.

We call this one variable x. An inequality is quadratic if there is a term which involves x^2 and no higher powers of x appear. Lower powers of x can appear.

Some examples of quadratic inequalities are:

  • x^2 + 7x -3 > 3x + 2
  • 2x^2 - 8 ≤ 5x^2
  • x + 7 < x^2 -3x + 1

Here the first and third are strict inequalities, and the second one is not. However, the procedure for solving the problem will be exactly the same for strict inequalities and inequalities that are not strict.

Solving Quadratic Inequalities

Solving a quadratic inequality requires a few steps:

  1. Rewrite the expression such that one side becomes 0.
  2. Replace the inequality sign with an equality sign.
  3. Solve the equality by finding the roots of the resulting quadratic function.
  4. Plot the parabola corresponding to the quadratic function.
  5. Determine the solution of the inequality.

We will use the first of the example inequalities of the previous section to illustrate how this procedure works. So we will have a look at the inequality x^2 + 7x -3 > 3x + 2.

1. Rewrite the expression such that one side becomes 0.

We will subtract 3x + 2 from both sides of the inequality sign. This leads to:

x^2 + 4x - 5 > 0

2. Replace the inequality sign with an equality sign.

x^2 + 4x - 5 = 0

3. Solve the equality by finding the roots of the resulting quadratic function.

There are several ways to find the roots of a quadratic formula. If you want to read more about this I suggest reading my article about how to find the roots of a quadratic formula. Here we will choose the factoring method, since this method suits this example very well. We see that -5 = 5*-1 and that 4 = 5 + -1. Therefore we have:

x^2 + 4x - 5 = (x+5)*(x-1) = 0

This works because (x+5)*(x-1) = x^2 +5x -x -5 = x^2 + 4x - 5. Now we know that the roots of this quadratic formula are -5 and 1.

4. Plot the parabola corresponding to the quadratic function.

Plot of the quadratic formula

Plot of the quadratic formula

You do not have to make an exact plot as I did here. A sketch will be sufficient to determine the solution. What is important is that you can easily determine for which values of x the graph is below zero, and for which it is above. Since this is an upward opening parabola we know that the graph is below zero in between the two roots we just found and it is above zero when x is smaller than the smallest root we found, or when x is larger than the largest root we found.

When you have done this a couple of times you will see that you do not need this sketch anymore. However, it is a good way to get a clear view on what you are doing and therefore it is recommended to make this sketch.

5. Determine the solution of the inequality.

Now we can determine the solution by looking at the graph we just plotted. Our inequality was x^2 +4x -5 > 0.

We know that in x = -5 and x = 1 the expression is equal to zero. We must have that the expression is larger than zero and therefore we need the regions left from the smallest root and right of the largest root. Our solution will then be:

x < -5 or x > 1

Make sure to write "or" and not "and" because then you would suggest that the solution would have to be an x that is both smaller than -5 and larger than 1 at the same time, which is of course impossible.

If instead we would have to solve x^2 +4x -5 < 0 we would have done the exact same until this step. Then our conclusion would be that x has to be in the region between the roots. This means:

-5 < x < 1

Here we have only one statement because we only have one region of the plot we want to describe.

Remember that a quadratic function does not always have two roots. It might happen that it has only one, or even zero roots. In that case we are still able to solve the inequality.

What If the Parabola Has No Roots?

In the case that the parabola does not have any roots there are two possibilities. Either it is an upwards opening parabola that lies entirely above the x-axis. Or it is a downwards opening parabola that lies entirely under the x-axis. Therefore the answer to the inequality will either be that it is satisfied for all possible x, or that there is no x such that the inequality is satisfied. In the first case every x is a solution, and in the second case there is no solution.

If the parabola has only one root we are basically in the same situation with the exception that there is exactly one x for which equality holds. So if we have an upwards opening parabola and it has to be larger than zero still every x is a solution except for the root, since there we have equality. This means that if we have a strict inequality the solution is all x, except for the root. If we do not have a strict inequality the solution is all x.

If the parabola has to be smaller than zero and we have strict inequality there is no solution, but if the inequality is not strict there is exactly one solution, which is the root itself. This is because there is equality in this point, and everywhere else the constraint is violated.

Analogously, for a downward opening parabola we have that still all x are a solution for a non-strict inequality, and all x except for the root when the inequality is strict. Now when we have a larger than constraint, there is still no solution, but when we have a larger than or equal to statement, the root is the only valid solution.

These situations might seem difficult, but this is where plotting the parabola can really help you to understand what to do.

how-to-find-the-roots-of-a-quadratic-function

In the picture, you see an example of an upward opening parabola that has one root in x=0. If we call the function f(x), we can have four inequalities:

  1. f(x) < 0
  2. f(x) ≤ 0
  3. f(x) > 0
  4. f(x) ≥ 0

Inequality 1 does not have a solution, since in the plot you see that everywhere the function is at least zero.

Inequality 2, however, has as solution x=0, since there the function is equal to zero, and inequality 2 is a non-strict inequality that allows equality.

Inequality 3 is satisfied everywhere except in x=0, because there equality holds.

Inequality 4 is satisfied for all x, so all x are a solution.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.