For American high school students who are getting ready for college and competitive tests such as the SAT, ACT, or AP Calculus, it is important to know mathematical concepts. One of them is greater than or equal to (≥), which is applied to compare two values, variables, or expressions. ≥ denotes that a number can be more than or just equal to another number. As an illustration, if a pupil scores ≥ 90 in a test, he or she got 90 or more. It is important to understand this concept when solving inequalities, a frequently encountered question on standardized tests. Learning inequalities not only assists with high school studies but also enhances problem-solving ability for college-level mathematics, providing students with a solid advantage in their studies.

What is Greater than or Equal to?

Greater than or equal to denotes that something is either greater than or equal to a number. Symbol “≥” stands for “greater than or equal to.” Consider the following scenario. The term y ≥ 19 denotes a value of y that is greater than or equal to 19.

Greater than or Equal to can also be represented on a number line. To show inequalities on a number line we use the following:

• Open circles are used for numbers that are less than or greater than (< or >).
• Closed circles are used for numbers that are less than or equal to and greater than or equal to (≤ or ≥).

The condition of greater than or equal to is not applicable for two known numbers. We can never say, 13 ≥ 4 because we know that 13 > 4. However, if there is an unknown variable, let’s say v and we need to find out its value then ≥ is a possible case. We can say, v ≥ 3 meaning v is greater than 3, but there’s a possibility that it can be 3.

Example: Let’s consider 2 variables a and b. If we need to say that a is greater than b, we write a > b. On the other hand, if we need to say that a is less than y, we write a < b. Similarly, we can show that a and b are equal by writing a = b.

There are times when we need to show two conditions like greater than or equal to. As mentioned above, greater than or equal to means anything is either greater than or equal to a number. The symbol “≥” stands for greater than or equal to.

So we if have to say x is greater than or equal to y we write it as, x ≥ y.

Uses of Greater than or Equal to

The uses of greater than or equal to are as follows:

• The greater than or equal to sign is mostly used to show inequalities between two entities.
• Linear Programming Problems make use of greater than or equal to sign to maximize functions.

How to Use Greater than or Equal to Symbol?

“Greater than,” “less than,” “greater than or equal to,” and “less than or equal to” are the 4 inequality symbols. Shorthand notation for comparing two quantities is represented by inequality symbols. The sentence “9 is greater than 4,” for example, can be written as 9>4. Consider each sign as a mouth that will always swallow the larger of the two numbers being compared. This is an effective method for remembering which number is higher.

Let’s use the example of you going on vacation to better understand it. Your journey will be less than 20 days. We use the < sign for ‘less than’ inequality. Furthermore, you are only allowed to bring 1000ml of apple juice on the plane for this journey. We use the ≤ symbol for inequalities like ‘less than or equal to.’ It is the opposite of greater than or equal to meaning not greater than or equal to. You also need to pack at least 3 pairs of clothes. So we can use the greater than or equal to sign here i.e. 3 ≥ clothes.

Solved Examples of Greater than or Equal to

Here are some solved examples of Greater than or Equal to for you to prepare for your exam.

Example 1: If x is greater than or equal to 25 and y is less than or equal to 40, then which one of the following is always correct?

1. x is greater than y
2. (y – x) is greater than 15
3. (y – x) is less than or equal to 15
4. (x – y) is greater than or equal to 65

Solution: Let’s first write down the comparison with the greater than or equal to operator.

x ≥ 25

y ≤ 40 or -y ≥ -40

Adding both inequalities,

(x – y) ≥ 25 – 40

(x – y) ≥ -15

(y – x) ≤ 15

The correct option is 3.

Example 2: What is the solution of x squared greater than or equal to 3?

Solution: We have

\( x^2≥3 \)

\( x^2-3≥0 \)

\( x^2-(\sqrt{3})^2≥0 \)

\( (x – \sqrt{3}) (x +\sqrt{3}) ≥0 \)

\( ⟹x∈(−∞,−\sqrt{3}]∪[\sqrt{3},∞) \)

Example 3: Find a solution to the linear inequality. y ≥ 4x – 5 (1, 1) b. (2, 1) c. (3, 4) d. (3, 0)

Solution: y ≥ 4x – 5

y – 4x ≥ – 5

We put the values from the options in place of x and y and verify the inequality.

x = 1 , y = 1

y – 4x = 1 – 4(1) = -3 ≥ – 5

Therefore the inequality condition is satisfied.

x = 2 , y = 1

y – 4x = 1 – 4(2) = -7 ≤ – 5

Therefore the inequality condition is not satisfied.

x = 3 , y = 4

y – 4x = 3 – 4(4) = -13 ≤ – 5

Therefore the inequality condition is not satisfied.

x = 3, y = 0

y – 4x = 0 – 4(3) = -12 ≤ – 5

Therefore the inequality condition is not satisfied.

Hence, the correct option is 1.

Conclusion

For college-bound American high school students and for competitive tests such as the SAT, ACT, or AP Calculus, mathematical concepts like greater than or equal to (≥) are fundamental. The symbol not only serves to compare values but also serves a key role in solving inequalities—a skill crucially tested on standardized tests. Understanding these topics reinforces problem-solving skills, which lay a firm groundwork for more advanced college-level math courses. Through the mastery of inequalities and other basic mathematical concepts, students gain an advantage that leads to future academic and career success.