Quadratic Inequalities and Set Notation

A quadratic inequality asks where a quadratic expression is positive, negative or possibly zero. For example, consider

This does not have just one or two answers. It asks for every value of $x$ for which the left-hand side is less than 3. Begin by putting the inequality into the standard form

where the box represents the inequality sign. Subtracting 3 from both sides gives

Now factorise:

The corresponding equation $(x-3)(x+1)=0$ has roots $x=3$ and $x=-1$. These roots divide the number line into three intervals:

They are possible boundaries of the answer, rather than the complete answer by themselves.

Deciding which intervals work

Between any two consecutive roots, the sign of a quadratic expression stays the same. It can change sign only when the graph reaches or crosses the $x$-axis. You can therefore test one convenient value from each interval.

For $(x-3)(x+1)$:

The question requires the expression to be negative, so the solution is

The roots are not included because the inequality is strict: at either root, the expression equals zero rather than being less than zero.

This also follows from the graph. The coefficient of $x^2$ is positive, so the parabola opens upwards. It lies below the $x$-axis between its roots and above the axis outside them. If the leading coefficient were negative, the parabola would open downwards and the positive and negative regions would be reversed.

Including or excluding the roots

The inequality sign determines whether the roots belong to the solution set:

• $f(x)<0$ or $f(x)>0$: exclude any root, because $f(x)=0$ there;
• $f(x)\leq0$ or $f(x)\geq0$: include any root, because equality is allowed.

For example,

has roots $-4$ and $2$. The product is non-negative outside the roots, so

Both endpoints are included. Notice that an outside solution consists of two separate intervals; it cannot be written as one chained inequality.

A reliable general method

To solve a quadratic inequality:

1. Rearrange it so that one side is zero.
2. Solve the corresponding quadratic equation to find its real roots. Depending on the quadratic, you might factorise, complete the square or use the quadratic formula.
3. Put the roots in numerical order and use them to split the number line into intervals.
4. Determine the sign of the quadratic in each interval, using test values or a correct sketch.
5. Select the interval or intervals required by the inequality.
6. Include roots only if equality is allowed.
7. Check a value from the proposed answer in the original inequality.

This interval-testing method remains dependable when a quick sketch is difficult or when the quadratic has a negative leading coefficient.

Special cases

A quadratic may have a repeated root, where the graph touches the $x$-axis but does not cross it. For example,

is true for every real value of $x$, because a square cannot be negative. By contrast, $(x-2)^2<0$ has no real solutions. Always consider the sign of the expression, rather than assuming that every pair of roots creates a region between them.

Writing the answer in set notation

Braces describe a set, and the colon means “such that”. Therefore

means “the set of values $x$ such that $x$ is greater than $-1$ and less than $3$”. It is the set-notation form of the first example.

For two separate intervals, use a union:

This means that $x$ belongs to the first set or the second set. If the permitted domain matters, state it explicitly; for example, $\{x\in\mathbb{R}:-1<x<3\}$ restricts $x$ to the real numbers, whereas a restriction to integers would produce only the integer members of the interval.