Inequalities

An inequality is a mathematical statement that looks exactly like an equation except that the equals sign is replaced with one of the five inequality symbols.

≠ not equal to

< less than

≤ less than or equal to

> greater than

≥ greater than or equal to

Imagine two friends of ages 9 and 12. Each of the five inequality symbols can be used to write a statement about their ages, depending on which aspect about their ages is to be emphasized.

9 ≠ 12 says that 9 is not equal to 12

9 < 12 says that 9 is less than 12

9 ≤ 12 says that 9 is less than or equal to 12

12 > 9 says that 12 is greater than 9

12 ≥ 9 says that 12 is greater than or equal to 9

Notice that 9 < 12 and 12 > 9 are two ways to represent the same relationship; when the 9 and 12 are reversed, the inequality is also reversed.

The difference in meaning of "greater than" and "greater than or equal to" is subtle. For example, let a represent the age of a person. How would an equality be written to show that the age limit for voting is 18?

The expression a > 18 means that the age has to be greater than 18. The expression a ≥ 18 means that the age either can be greater than 18 or it can be equal to 18. Since a person who is exactly 18 years old is allowed to vote, the inequality a ≥ 18 is the correct one to use for this situation.

Various limits in society can be expressed using inequalities. Just a few examples are speed limits, minimum and maximum bank withdrawals, minimum and maximum fluid levels, grade requirements for college admittance, and minimum sales to make a fundraising goal. Some common phrases that indicate inequality within word problems are "minimum," "maximum," "at most," "at least," and superlatives like "oldest" and "smallest."

Solving Inequalities

Every possible equation can be made into an inequality. Furthermore, inequalities are solved the same way as equations, with one exception; namely, when an inequality is multiplied or divided by a negative number on both sides of the inequality sign.

For example, start with the true statement 5 < 6. When both sides are multiplied by −2, the statement becomes −10 < −12. This resulting statement is false because −10 is greater than −12. Since multiplying (or dividing) an inequality by a negative number results in a false statement, the inequality symbol must be reversed to maintain a true statement. In this example, the correct answer is −10 > −12. As another example, to solve −4x > 12, divide both sides by −4, and then reverse the inequality to yield x < −3.

A Practical Example. Consider the following problem. A student has saved $40, and wants to save a total of at least $300 to buy a new bicycle and bicycle gear. His job pays $7.25 an hour. Find the number of hours he needs to work in order to save at least $300.

To solve this problem, let h be the number of hours he needs to work. The words "at least" indicate that the combination of savings and pay must be greater than or equal to $300. The solution below shows that the student must work 36 or more hours to meet his goal of saving $300.

$40 + 7.25h ≤ $300

7.25h ≤ 260: Subtract 40 from each side.

h ≥ 35.9: Divide each side by 7.25.

Graphing Inequalities

The number lines in the top five rows of the boxed figure illustrate each case that may arise in graphing inequalities in one variable, x. A closed circle on a number indicates that it is included in the solution set, and an open circle indicates that it is not included in the solution set. Shading to the left with a darkened arrow on the end indicates all numbers of lesser value are included in the solution set; shading to the right with a darkened arrow on the end indicates all numbers of greater value are included in the solution set.

Graphing Compound Inequalities. Compound inequalities are two inequalities separated by the words "and" or "or." The solution set to a compound inequality that is separated by the word "and" is the region where the two graphs overlap. This is known as an intersection.

Consider how to write a compound inequality that represents the possible ages of a teenager, and then how to graph the solution set. To write this inequality, let a represent age. So a ≥ 13 and a < 20. The graph of the intersection is shown in the sixth number line in the boxed figure.

Compound inequalities that represent solutions that fall between two numbers are frequently written in an abbreviated notation with the variable in the middle. Hence, the inequality described above can also be written as 13 ≤ a < 20.

The solution set to a compound inequality that is separated by the word "or" is the combination of all points on both graphs. This is known as a union.

Consider how to write a compound inequality to represent the possible ages of a sibling who is not a teenager, and then how to graph the solution set.

To write this inequality, let a represent age. So a < 13 or a ≥ 20. The graph of the union is shown in the seventh number line in the boxed figure.

The corollary to compound inequalities in one variable is a system of inequalities in two variables. The solution to a system of inequalities is the intersection of the two graphs, just like the solution to a system of equations. However, the intersection of a system of inequalities usually consists of a whole region of points. This is especially useful since real-world problems often involve choosing from several possible solutions.

Inequalities in Two Variables

The graph below shows how to graph a linear inequality in two variables. Graphs in two variables are drawn using either a solid or dashed line. A solid line indicates that the points on the line are included in the solution set, and a dashed line indicates that the points on the line are not included in the solution set. For a linear graph, the solid or dashed line divides the plane into two regions, only one of which will be the solution set.

Consider how to solve and graph y + 7 > 3x + 8. First, subtract 7 from both sides to obtain y > 3x 1. Graph the equation y = 3x + 1 using a dashed line. The dashed line divides the plane into two regions. Choose one point from each region that will be easy to substitute into the inequality. The points chosen below are (0, 0) and (0, 2). Only one of the two points will be a solution to the inequality, indicating which region includes the set of solutions.

First, substitute (0, 0).

y > 3x 1

0 > (3 × 0) + 1

0 > 1 (FALSE)

Next, substitute (0, 2).

y > 3x + 1

2 > (3 × 0) + 1

2 > 1 (TRUE)

Hence, the point (0, 2) is a solution to the inequality. If more points in the same region are tested, they will also be solutions. Hence, the graph of the solution set includes points to the left of, but not including, the line y = 3x + 1, as shown above.

see also Functions and Equations.

Michelle R. Michael

Bibliography

Dantzig, George B. Linear Programming and Extensions. Princeton, NJ: Princeton University Press, 1998.

SUMMARIZING INEQUALITIES

Two points must be remembered when dealing with inequalities.

• If each side of an inequality is multiplied or divided by a positive number, then no changes are made to the inequality symbol.
• If each side of an inequality is multiplied or divided by a negative number, then the equality symbol must be reversed.