An inequality is simply a comparison between two values or expressions. It tells you if one side is bigger or smaller than the other. For example, a simple statement like "5 is greater than 3" is an inequality.
Inequalities are really useful in math, and you use them in everyday life. For example, when you're shopping and trying to stick to a budget, you're using inequalities. If you want to buy a shirt that costs $20 but you only have $15, you can say "the cost of the shirt is greater than the amount of money I have" (20 > 15).
So knowing about inequalities is pretty handy, and it can help you make better decisions and understand the world around you. It’s also useful if you want to journey further into more complex mathematics in the future.
The Origin of Inequality in Math
Inequalities have a long and storied history in mathematics, dating back to ancient civilizations. Inequalities have been around since Mesopotamia and Egypt, where they were used to figure out how to measure land and build things.
In ancient Greece, around 300 BC, the philosopher Euclid used geometric inequalities to prove his geometric theorems in his work, The Elements. In his book Arithmetica, which was written in 250 AD, the Greek mathematician Diophantus also made important contributions to the idea.
The Italian mathematician Niccol Fontana Tartaglia (1499-1557) and the French mathematician François Viète (1540-1603) pioneered the use of algebraic methods to solve inequalities during the Renaissance.
Inequalities were studied more in depth by mathematicians like Pierre de Fermat, René Descartes, and Johann Bernoulli in the 17th and 18th centuries. These mathematicians laid the groundwork for how inequalities are studied today.
Then, in the 1800s, the study of inequalities was taken up by notable mathematicians such as Joseph-Louis Lagrange and Augustin-Louis Cauchy. They developed many of the techniques and concepts that are still used today.
Understanding Inequality Symbols
There are four basic symbols used for inequalities:
- Less than (<): This symbol is used to show that one value is strictly smaller than another. For example, the inequality 3 < 5 means that 3 is less than 5, but not equal to 5.
- Greater than (>): This symbol is used to show that one value is strictly larger than another. For example, the inequality 7 > 5 means that 7 is greater than 5, but not equal to 5.
- Less than or equal to (≤): This symbol is used to show that one value is less than or equal to another. For example, the inequality 3 ≤ 5 means that 3 is less than or equal to 5.
- Greater than or equal to (≥): This symbol is used to show that one value is greater than or equal to another. For example, the inequality 7 ≥ 5 means that 7 is greater than or equal to 5.
It's vital to understand the difference between strict inequalities (<, >) and non-strict inequalities (≤, >) because they have different meanings and can lead to different results when solving problems.
Also, it's worth noting that these symbols can be used to compare not only numbers but also expressions.
For example: x+3 < y+4 or x^2+5x+6 > 0
It's important to be familiar with the properties of these expressions and the operations you can perform on them to solve the inequality.
Types of Inequalities
There are several types of inequalities you’ll come across, depending on how far you delve into the complex arena of mathematics. However, these are the most common ones:
Linear Inequalities
These are linear function inequalities (i.e., ax + b, where a and b are constants).
Example: 2x + 3 < 7.
Quadratic Inequalities
These use quadratic functions. (i.e., ax^2 + bx + c, where a, b, and c are constants).
Example: x^2 - 4x + 3 > 0.
Absolute Value Inequalities
This inequality has one side set as an absolute value.
Example: |x - 3| < 5.
Interval Notation
Interval notation, denoted by "[]," is a method of expressing a range of values using inequalities.
Example: [2, 5], where the set of all real numbers is greater than or equal to 2 and less than or equal to 5.
Piecewise-Defined Inequalities
Inequalities that have different parts are called piecewise-defined inequalities, and each expression uses a different inequality sign.
Example: f(x) =
{ 2x - 1 if x < 3}
{ 5 if x ≥ 3 }
Rational Inequalities
Rational inequalities are expressed as a ratio of polynomials.
Example (x^2 - 4x + 3)/(x - 2) < 0.
Complex Inequalities
If you see an inequality with the term i, that’s a complex inequality.
Example: (3 + 4i) < (2 + 5i).
These are just a few examples of the types of inequalities that exist in mathematics. Inequalities can be solved in different ways, and it's important to know the differences between them if you want to get good at them.
Rules for Inequalities
There are several rules that apply to inequalities that can help you quickly solve some math problems. Let’s go over some basic ones here.
1. Addition or subtraction
When you add or subtract the same value from both sides of an inequality, the inequality remains true.
Examples
- x + 3 < 7 is equal to x < 4
- x - 8 > 10 is equal x > 18
2. Multiplication or division
When you multiply or divide both sides of an inequality by a positive number, the inequality remains true. However, if you multiply or divide both sides by a negative number, you must reverse the inequality symbol.
Examples
- x < 4 is equal to 2x < 8
- x > 7 is equal to x/2 > 3.5
- -x > 4 is equal to x < -4
3. Transitive property
If a less than b and b less than c, then a is less than c.
Example
- If x < y < z, then x < z
4. Interval notation
To express the solution of an inequality in interval notation, you have to find the values that make the inequality true and then use brackets or parentheses to indicate if the number is included or not in the solution set.
An example: If the solution is x > 3, then the interval notation is (3,+∞), because the solution is all real numbers greater than 3. If the solution is x ≥ 3 then the interval notation is [3,+∞) because 3 is included as part of the answer.
5. Solving compound inequalities
When solving a compound inequality, you have to first solve each inequality separately and then find the common solution. For example, if x < 3 or x > 5, then the solution is x < 3 or x > 5.
Example: Solve the compound inequality: 3x + 2 < 7 or x - 4 > 6
Solution:
1. First, solve each inequality separately:
3x + 2 < 7 can be simplified to x < 2
x - 4 > 6 can be simplified to x > 10
2. Now, find the common solution between the two inequalities:
The solution of x < 2 doesn't overlap with the solution of x > 10, so the solution of the compound inequality is the union of the two solutions:
x < 2 or x > 10
3. You can use interval notation to express this solution where extremes are not included in the solution, like this:
(-∞,2) U (10,+∞)
Keep in mind that these rules apply to most of the inequalities, but there are some exceptions and special cases, you should always check the domain of the inequality and be aware of the properties of the expressions used in the inequality to ensure that the solution found is correct.
Real-World Use for Inequalities
Of course, you might be thinking, “How does this help me in life?” Inequalities are all around us, and you use them throughout the day even if you don’t realize it. For example, when you're trying to decide if you should bring an umbrella or not, you're using inequalities. You're thinking something like, "if the chance of rain is greater than 50%, I'll bring an umbrella." (Chance of rain > 50%).
Or think about when you're trying to decide whether to buy that new iPhone or not– that’s an inequality too. Your thought process might be like "if the iPhone’s price is less than my budget, I'll buy it" (iPhone < your budget).
So don’t discount the power of inequalities. Learning this skill not only makes you a better mathematician, but it also helps with understanding the world we live in and making more informed choices.
Hire a Math Tutor Today
Now that you have a better understanding of inequalities as a concept, it’s time to learn how to put it in action. Learner’s experienced math tutors can guide your child in all math subjects, from trigonometry to calculus.
Inequalities pop up all through math, so it’s an important skill to get right and learn how to apply the rules to solve complex problems in the future. Contact Learner today to match you or your child with a private math tutor.
