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In the vast landscape of mathematical symbols, the** “greater than or equal to”** sign **(≥)** stands as a powerful tool to denote inclusivity and range. When transposed onto a number line, this seemingly simple concept takes on a visual dimension, making abstract comparisons tangible and easy to comprehend.

In this article, we will embark on a journey through the realm of **≥,** exploring its nuances and significance as it manifests on the number line.

**Defining Greater Than or Equal to on a Number Line**

On a number line, the symbol **“greater than or equal to”** **(denoted as ****≥****)** signifies that a number, located at a particular point, is either greater than another specific number or exactly equal to it.

In visual terms, when a value $a$ is** “greater than or equal to”** a value $b$, the point representing $a$ lies to the right of or coincides with the point for $b$. Often, this relationship is indicated with a closed circle at the point representing $b$ and a line or arrow extending to the right, showing that all numbers in that direction, including $b$, satisfy the condition **$a≥b$****.** This visual representation aids in quickly grasping the relative positions and values of numbers in relation to one another.

**Example**

Solve the compound inequality: $2≤x+1≤5$

### Solution

Subtract 1 from all parts: $1≤x≤4$

On a number line, this is represented by a closed dot at 1, a closed dot at 4, and a solid line connecting them, indicating all numbers in the inclusive range [1, 4].

Figure-1.

**Properties**

The **“greater than or equal to”** symbol, represented as **≥,** is a partial order relation on the real numbers. When visualized on a number line, this relation and its properties become evident. Let’s delve into these properties in the context of the number line:

**Reflexivity:**- Every number is
**greater than or equal to**itself. Mathematically, for any number $a$,**$a≥a$**. On a number line, this can be visualized as a point lying on itself.

- Every number is
**Antisymmetry:**- If $a$ is
**greater than or equal to**$b$ and $b$ is**greater than or equal to**$a$, then $a$ must be equal to $b$. Mathematically, if $a≥b$ and**$b≥a$,**then $a=b$. This means that on a number line, the points representing $a$ and $b$ coincide.

- If $a$ is
**Transitivity:**- If $a$ is
**greater than or equal to**$b$, and $b$ is**greater than or equal to**$c$, then $a$ is also**greater than or equal to**$c$. Mathematically, if**$a≥b$**and**$b≥c$**, then**$a≥c$**. On a number line, if $a$ lies to the right of $b$ and $b$ lies to the right of $c$, then $a$ will also lie to the right of $c$.

- If $a$ is
**Addition and Subtraction Property:**- If
**$a≥b$**, then**$a+c≥b+c$**and**$a−c≥b−c$**for any real number $c$. When you add or subtract a constant value to both numbers, their relative positions on the number line with respect to**“greater than or equal to”**don’t change.

- If
**Multiplication Property:**- If
**$a≥b$**and $c>0$, then $ac≥bc$. If you multiply both sides of an inequality by a positive number, the inequality direction remains unchanged. - However, if
**$a≥b$**and $c<0$, then $ac≤bc$. If you multiply both sides of an inequality by a negative number, the direction of the inequality flips. On a number line, this means the order of $a$ and $b$ will switch with respect to the origin when multiplied by a negative value.

- If
**Density Property:**- For any two distinct real numbers $a$ and $b$ where $a<b$, there exists another real number $c$ such that $a<c<b$. This means that between any two points on the number line, there are infinitely many other points.

**Exercise**

**Example 1**

Solve the inequality: **x + 3 ≥ 7**

**Solution**

Subtract 3 from both sides: **x ≥ 4**

On a number line, this would be represented with a closed dot at 4 and a line or arrow extending to the right **(indicating all numbers greater than or equal to 4)**.

Figure-2.

**Example 2**

Solve the inequality: **2x – 5 ≥ 9**

**Solution**

Add 5 to both sides: **2x ≥ 14**

Divide by 2: **x ≥ 7**

On a number line, a closed dot at 7 with a line or arrow extending to the right.

Figure-3.

**Example 3**

Solve the inequality: **-3x + 2 ≥ 11**

**Solution**

Subtract 2 from both sides:** -3x ≥ 9**

Divide by -3 (Remember to reverse the inequality when dividing by a negative number): x ≤ -3

On a number line, a closed dot at -3 with a line or arrow extending to the left.

**Applications **

The **“greater than or equal to” (≥)** relation, especially when visualized on a number line, serves various functions across multiple fields. Here’s how this relation is applied across different disciplines:

**Mathematics****Algebra:**Inequalities are commonly used to solve problems involving limits, such as determining the range of possible values of a variable.**Calculus:**In optimization problems, inequalities help determine the maximum or minimum values of functions within a given domain.

**Economics and Finance****Budgeting:**If a company’s earnings are**greater than or equal**to its expenditures, it is profitable or breaking even.**Stock Analysis:**Investors might use inequalities to set benchmark performances. For example, they might look for stocks with returns**≥ 8%.****Linear Programming:**This mathematical modeling technique uses inequalities to determine the best outcome (such as maximum profit or minimum cost) in a given mathematical model.

**Computer Science****Algorithms:**Many algorithms, especially sorting and searching ones, use the**≥**relation for making decisions.**Database Queries:**When filtering or searching through large datasets, queries often involve**≥**conditions to retrieve relevant records.

**Physical Sciences****Physics:**Inequalities might be used to set bounds on measurements, like ensuring a particle’s speed is less than or equal to the speed of light.**Chemistry:**In equilibrium reactions, chemists might look for conditions where the concentration of one substance is**greater than or equal**to another.

**Engineering****Safety Standards:**Structures are often designed to withstand loads**greater than or equal**to the maximum expected load to ensure safety.**Signal Processing:**Filters might be designed to allow frequencies**greater than or equal**to a certain threshold to pass through.

**Medicine and Biology****Epidemiology:**Researchers might track if the number of cases of a disease is**greater than or equal to**a threshold that requires intervention.**Pharmacology:**Dosages might be recommended if a patient’s weight is**greater than or equal to**a certain number.

*All images were created with GeoGebra.*