Linear inequality and geometric interpretation

Inequalities are more flexible than equations because they allow you to model realistic problems and have a more general graphical representation.

Although the most common thing is to deal with problems that only involve equations, the truth is that inequalities arise naturally in various contexts:

• If a medication requires storage between $15º \ C$ and $30º \ C$, its temperature $T$ should satisfy the inequalities $T \geq 15$ and $T \leq 30$. In this case, it could be important to know the corresponding temperature range in Fahrenheit.
• Scales (like any measuring instrument) have a margin of error of $\pm e$ grams. This means that if an object was weighed and the scale indicated a value of $25$ grams, then (due to the error) its true weight would be in the range of $25 \pm e$. So, the true weight $p$ should satisfy both $p \geq 25 - e$ and $p \leq 25 +e$. If the margin of error was equal to $5$, then the true weight would be between $20$ and $30$.
• In finance, the value at risk $VaR$ is the maximum loss that a company expects to have within a period of time with a confidence of $95\%$. This means that it expects that within this period of time, $95\%$ of the losses will be less than $VaR$. This value is a pivotal measure of risk for companies — being able to calculate it accurately allows them to have an idea of how much money they can afford to lose in the future and not just that: knowing this, they can then generate strategies to protect themselves from risk.

In this article, you will learn to interpret and visualize linear inequalities on the number line, on the plane, and in space.

One dimensional space

When a number $x$ is less than another number $y$ we use the notation $x < y$. Similarly, $x > y$ indicates that $x$ is greater than $y$. Some examples would be $2 < 3$ and $10 > 6$. In these cases, we speak of a strict inequality because $x$ cannot be equal to $y$.

However, in many situations, it is convenient to leave open the possibility that both numbers are equal. In this way, the notation $x \leq y$ means that $x < y$ or that $x = y$. In turn, $x \geq y$ means that $x > y$ or that $x = y$. In these situations, you are dealing with a non-strict inequality. Furthermore, since the disjunction is not exclusive, it always holds that $x \leq x$. Some concrete examples would be $1 ≥ -4$ and $7 ≤ 7$.

As you know, solving an equation containing a variable consists of finding the value of the variable that makes the equation true. For example, to solve the equation $x + 4 = 9$ means to find such a number $x$ that when adding $4$ to it, the result is $9$. On the other hand, solving an inequality consists of finding all the values that make said inequality true. This is often more complicated than solving an equation, and it is best to find a way to compactly describe all possible solutions.

Suppose you want to solve the inequality $x < 3$. You have to determine those numbers that are less than $3$ and you can think of some options like $1$, $-5$, or $-17$. At this point, it is clear that the number of solutions is infinite and so it would be a good idea to have a graphical description of the set of all possible solutions. If you start by drawing the number line and highlighting the number $3$, then the solution set consists of all points to the left of (of course, not including) $3$. You call this set a ray and denote it as $(-\infty, 3)$.

Now, if you were to use the inequality $x \leq 3$, the solutions would be all the numbers except $3$ along with $3$ itself. Graphically you would have a ray that starts at $3$ (including it) and extends to its left along the entire number line. Denote this new set of solutions as $(-\infty, 3]$.

But, if you were treating the inequality $x > 3$, then any number greater than $3$ would be valid, so the solution set would be the ray starting at (but not including) $3$ and extending to its right, along the entire number line, which is denoted by $(3, \infty)$.

Clearly, the solutions of the inequality $x \geq 3$ form a similar ray, only now the $3$ is included and you denote it by $[3, \infty)$.

As you can see, in the notation of the rays the parentheses mean that the number is not included, while the bracket indicates that the number is included.

An important feature of inequalities is that they can be combined. If you have the inequalities $2 < x$ and $x \leq 6$, then you know that the solutions of the first one are the ray that starts at $2$ (not including it) and extends to its right, while the solutions of the second inequality form the ray that starts at $6$ (including it) and extends to its left. Combining both inequalities would imply thinking of a set of numbers that satisfies both simultaneously. If you graph the two solution rays, then the numbers that simultaneously satisfy both inequalities would be precisely those that are at the intersection of both rays.

When you have two combined inequalities, the notation of their solution set depends on the type of inequalities involved. For two inequalities $a<x$ and $x<b$, we express their combination using the notation $a < x < b$ and denote the set of solutions as $(a, b)$ and call it an interval. There are different combinations for the inequalities involved, so you need to list all possible cases along with their notation and solution set in the following table:

Two-dimensional space

You know that a linear equation represents a line in the plane and has the form:$$ax + by = c$$Linear inequalities are similar and have one of the following aspects:

$$ax+by <c \qquad ax+by \leq c \qquad ax+by > c \qquad ax+by \geq c$$

To find the solutions of a linear equation, one has to find all the points on the line it describes. On the other hand, linear inequalities produce a more general region of solutions. To find it, the first thing you have to do is convert the inequality into an equality (replacing the inequality sign with a $=$) and determine the generated line. This line separates the plane into two regions: the one above it (upper semi-plane) and the one below it (lower semi-plane). Then the region of solutions will be one of the half-planes, which depends on the type of inequality.

Also, if the inequality is strict, then the line is not included in the set of solutions, while if the inequality is not strict, the line is included.

Since the solution set is one of these two regions, it is enough to choose one and take a test point within it: if the point satisfies the inequality, then the correct solution region is the chosen one; otherwise, you can be sure that the solution set is the other region.

Suppose you have the inequality $3x + 2y < 6$. To find the solution set, first consider the line $3x + 2y = 6$ and plot it. In order to do this, you can isolate $y = 3 - \frac{3}{2}x$, from the region where you see that the line passes through the point $(0, 3)$ and has slope $-\frac{3}{2}$. Now identify the upper and lower regions. As a test point, let's choose the origin $(0,0)$ which is in the lower region. You see that $3 \cdot 0 + 2 \cdot0 = 0 < 6$. Since a point in the lower region satisfies the inequality, we are sure that this is the solution region, and even better, because the inequality is strict, the region does not contain the line.

Same as you did with the inequalities on the number line, you can combine the inequalities on the plane as well. In this case, you must determine the solution region of each inequality and then the set of simultaneous solutions will be the intersection of these regions.

Let's illustrate this with an example. Suppose a small publisher issues books on both Philosophy and History. Each year they must:

1. not publish more than 50 books;
2. publish at least 10 books on Philosophy;
3. publish at least as many books on History as on Philosophy.

Find the set of all combinations of numbers of Philosophy and History books that the publisher can issue to satisfy their obligations.

If you denote by $x$ and $y$ the number of published books on Philosophy and History, respectively, you get three inequalities:

$$x + y \leq 50, \qquad x \geq 10, \qquad y > x$$Let’s graph the solutions of these three inequalities and then intersect all the regions. Note that the second inequality consists of the region to the right of the vertical line that cuts the X-axis at 10, while the region of the third inequality consists of those points that lie above the identity line $y = x$. The line associated with $x + y ≤ 50$ is given by $y = 50 -x$, so it intersects the $Y$-axis at $50$, and has a slope of $-1$. Also, testing the origin, you can see that $0 + 0 = 0 \leq 50$, so the solution region is the lower half-plane that includes the line.

Now let's intersect the three regions obtained to find the solution region:

Three- and N-dimensional space

In three-dimensional space, a linear equation looks like the following:

$$ax + by + cz = d$$This equation defines a plane in space.

A linear inequality in the space would be for example $$ax + by + cz < d$$It separates the space into two regions: the one above the plane $ax + by + cz = d$, and the one below it. These sets are called upper and lower half-spaces, respectively.

The strategy for solving linear inequalities in space is the same as in the plane. Also, as in the two-dimensional case, in space, you can consider simultaneous inequalities and the solution region is then the intersection of all the half-spaces

As you can see, working in three-dimensional space is very similar to working in the plane: although the graphs are a bit more complicated, all the algebraic properties are the same. This gives us a hint of how to generalize our work to $N-$dimensional space. Linear equations would look like this:

$$a_1x_1 + a_2 x_2 + \dots + a_n x_n = b$$The solution set is called a hyperplane, and although we cannot graph it, its algebraic properties are analogous to those of lines and planes. A linear inequality can be like $$a_1x_1 + a_2 x_2 + \dots + a_n x_n < b$$and it separates the $N$-dimensional space into two regions, one above the hyperplane and one below it. Although we don't have a visual representation, you can use the same techniques as in the spaces of lower dimension.

Conclusion

• Solutions of linear inequalities can be represented geometrically in a simple way.
• In the one-dimensional space, the linear inequalities generate intervals and rays.
• In the plane, the linear inequalities are like $ax + by < c$, and generate a line that separates the plane into $2$ regions: one above it and one below it. The solution set is one of these half-planes and can be easily determined with a test point.
• In three-dimensional space, the linear inequalities are like $ax + by + cz< d$, but the situation is similar to the two-dimensional case, because now there is a plane that separates space into two regions.
• For $N$−dimensional space the linear inequalities are like $a_1x_1 + a_2 x_2 + \dots + a_n x_n < b$ and although we don't have a graphical representation, the algebraic techniques are the same.