Less Than or Equal To Sign

• Introduction
• Definition of Less Than or Equal To
• Symbol of Less Than or Equal To
• Greater Than or Equal To and Less Than or Equal To
• Less Than or Equal To on a Number Line
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Introduction

Less than or equal to is a mathematical concept used to compare two quantities. It tells us whether one quantity is smaller or equal to another. Imagine a number line where numbers to the left are smaller and numbers to the right are larger. When we use `≤`, it means we're including the number itself along with all the smaller numbers. For example, if we say `x ≤ 5`, it means that `x` is either equal to `5` or less than `5`.

Inequality symbols are very helpful in maths while defining the domain and range of a graph, writing inequality statements to represent a real-world scenario, setting boundaries for the variable(s) in a function, etc.

Definition of Less Than or Equal To

Less than or equal to `(≤)` is a concept in math that tells us if one thing is either smaller or the same as another. It's used in various phrases like "at most," "no more than," "a maximum of," or "not exceeding." For instance, if we have the inequality `2x - 3 ≤ 9`, it means that twice a number `(x)` minus three is not more than nine. Simplifying, we find `x ≤ 6`, indicating that the number `(x)` can be six or less than `6`. This concept allows us to explore values that are either smaller or equal in mathematical expressions.

Symbol of Less Than or Equal To 

The symbol `≤` represents "less than or equal to". This is a closed inequality symbol and is a combination of less than sign and equal to sign. If an inequality symbol contains the equal to sign below the inequality symbol, we call it a closed inequality. However, if an inequality symbol does not contain the equal to sign below the inequality symbol (like `<` or `>`) we call it an open inequality. 

Let's understand the usage of the symbol `≤` with a new scenario. Consider a marathon runner, Sarah, training for a race. She plans to run a maximum of `10` miles each day. What does "maximum" imply in this context? It means Sarah will run either less than or equal to `10` miles daily. If we denote the number of miles Sarah runs as `y`, we can express this mathematically as `y ≤ 10`.

Greater Than or Equal To and Less Than or Equal To

We often confuse between the symbols `≤` (less than or equal to) `≥` (greater than or equal to). The symbol `≥` represents an inequality opposite to "less than or equal to", called "greater than or equal to". It signifies that one value is either greater than or equal to another. In mathematics, we encounter statements involving both `≤` and `≥` signs, known as inequalities, each sign carrying a distinct meaning. We can understand their differences through comparison. Here, we'll present some examples illustrating these symbols and their respective meanings.

Less Than or Equal To on a Number Line

The concept of "less than or equal to" on a number line is fundamental in mathematics for comparing numerical values. It indicates that one number is either smaller than or equal to another. On the number line, this relationship is depicted by placing a closed circle `(●)` on the point corresponding to the smaller or equal value. For instance, `x ≤ 1` is shown this way on the number line. 

The closed circle emphasizes that the endpoint is included in the comparison.This visual representation helps understand the relative magnitudes of the values being compared.

Solved Examples

Example `1`: A library has a capacity of `100` books on its shelves. Write an inequality using the less than or equal to symbol to represent this statement.

Solution:

Let's represent the number of books by the variable `y`.

It is given that a library has a capacity of `100` books on its shelves.

So, this means the library can hold no more than `100` books on its selves.

Therefore, `\color{#38761d}y \color{#6F2DBD}≤ \color{#fb8500}100`.

Example `2`: Solve the inequality \( 2y - 5 \leq 7 \).

Solution:

To solve the inequality \(2y - 5 \leq 7\), we follow these steps:

• Add \(5\) to both sides of the inequality:

\(2y - 5 + 5 \leq 7 + 5\)

\(2y \leq 12\)

• Divide both sides by \(2\) to isolate \(y\):

\(\frac{2y}{2} \leq \frac{12}{2}\)

\(y \leq 6\)

So, the solution to the inequality is \(y \leq 6\). This means that \(y\) can take any value less than or equal to \(6\) that satisfies the inequality.

Example `3`: A bus can carry a maximum of `50` passengers. Express this limitation using the less than or equal to symbol.

Solution:

Let's represent the number of passengers by the variable `x`.

It is given that the bus can carry a maximum of `50` passengers.

So, this can be represented by a less than or equal to inequality.

Therefore, `\color{#38761d}x \color{#6F2DBD}≤ \color{#fb8500}50`.

Example `4`:  If \( x + 3 \leq 8 \), what are the possible values of \( x \)?

Solution:

To solve the inequality \( x + 3 \leq 8 \), we follow these steps:

• Subtract \(3\) from both sides of the inequality:

\(x + 3 - 3 \leq 8 - 3\)

\(x \leq 5\)

So, the possible values for \(x\) are any number less than or equal to \(5\).

Example `5`: A rectangular field has a length of \( 10 \) meters. Write and solve an inequality to find the possible values of the width of the field if its perimeter must be less than or equal to \( 40 \) meters.

Solution:

Given a rectangular field with a length of \(10\) meters.

Let \(w\) represent its width. 

Formula for perimeter \(P\) of a rectangle is,

\(P = 2 \times (\text{length} + \text{width})\)

Substitute the value of length in the formula.

\(P = 2 \times (10 + w)\)

\(P = 20 + 2w\)

The inequality \(20 + 2w \leq 40\) represents the condition that the perimeter must be less than or equal to \(40\) meters. 

To solve the inequality \(20 + 2w \leq 40\), we follow these steps:

• Subtract \(20\) from both sides of the inequality:

\(20 + 2w - 20 \leq 40 -20\)

\(2w \leq 20\)

• Divide both sides by \(2\) to isolate \(w\):

\(\frac{2w}{2} \leq \frac{20}{2}\)

\(w \leq 10\)

Therefore, the width of the field can be any value less than or equal to \(10\) meters.

Example `6`: Identify the inequality based on the given number line.

Solution:

Here in the number line, the arrow is pointed towards the left which indicates the values are less than `3`.

And the closed circle `(●)` on the point indicates that the value `3` is included in the inequality.

So the inequality for the given number line is \(x \leq 3\).

Example `7`: Determine the inequality based on the given number line.

Solution:

Here in the number line, the arrow is pointed towards the right which indicates the values are greater than `-1`.

And the closed circle `(●)` on the point indicates that the value `-1` is included in the inequality.

So the inequality for the given number line is \(x \geq -1\).

Practice Problems

Q`1`. Solve the inequality \(3x + 2 \leq 11\).

1. \(x \leq 3\)
2. \(x \geq 6\)
3. \(x \leq 6\)
4. \(x \geq 3\)

Answer: a

Q`2`. Determine the inequality based on the given number line.

1. \(y \leq 1\)
2. \(y \geq 6\)
3. \(y \leq 6\)
4. \(y \geq 1\)

Answer: c

Q`3`. A swimming pool has a capacity of `150,000` liters of water. Write and solve an inequality to find how many liters of water can be stored in the pool at most.

1. \(x \geq 100,000\)
2. \(x \geq 150,000\)
3. \(x \leq 140,000\)
4. \(x \leq 150,000\)

Answer: d

Q`4`. Identify the inequality based on the given number line.

1. \(t \geq 1\)
2. \(t \leq -4\)
3. \(t \leq 1\)
4. \(t \geq -4\)

Answer: d

Q`5`. Solve the inequality \(2w + 4 \leq 12\).

1. \(w \leq 3\)
2. \(w \leq 4\)
3. \(w \leq 16\)
4. \(w \leq 8\)

Answer: b

Frequently Asked Questions

Q`1`. What is an inequality?

Answer: An inequality is a mathematical statement that compares the relative size of two expressions, typically using symbols like `<` (less than), `>` (greater than), `≤` (less than or equal to), or `≥` (greater than or equal to).

Q`2`. What is the Difference Between Less Than and Less Than or Equal To?

Answer: "Less than" `(<)` indicates that one value is strictly smaller than another, while "less than or equal to" `(≤)` includes the possibility of equality, allowing for the comparison of values that are equal or smaller.

Q`3`. What does "less than or equal to" mean?

Answer: "Less than or equal to," denoted as `≤`, signifies that one value is either smaller than or equal to another. For example, \(x \leq 8\) means that \(x\) can be any number less than or equal to `8`.

Q`4`. What is the difference between an equation and an inequality?

Answer: An equation indicates that two expressions are equal, while an inequality indicates a relationship between two expressions that may not be equal. Inequalities involve symbols such as `<`, `>`, `≤`, or `≥`, whereas equations typically involve an equal to sign `(=)`.

Q`5`. How are inequalities used in real life?

Answer: Inequalities are used in various real-life scenarios, such as budgeting (where expenses must be less than or equal to income), scheduling (where time constraints apply), and resource management (where quantities must not exceed certain limits). They help in decision-making processes by setting boundaries or constraints on variables.