What Are Linear Inequalities – Meaning, Properties & Examples

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Algebraic expressions are not only used to form equations such as $2x – 5 = 3x + 7$, but also inequalities. Inequalities are also formed by algebraic expressions having inequality signs like $\lt$, $\gt$, $\le$, or $\ge$.

Let’s understand what are linear inequalities and how to solve them with examples.

What are Linear Inequalities?

Inequalities are the mathematical statement that shows the relation between two expressions using the inequality symbol. The expressions on both sides of an inequality sign are not equal. It means that the expression on the left-hand side should be greater than or less than the expression on the right-hand side or vice versa. If the relationship between two algebraic expressions is defined using the inequality symbols, then it is called literal inequalities. When the algebraic expression present is of degree $1$, then such inequalities are called linear inequalities.

Examples of Linear Inequalities

Example 1: $2x – 3 \lt 9$

The LHS of the inequality is $2x – 3$

The RHS of the inequality is $9$

The inequality means $2x – 3$ is less than $9$.

Example 2: $7y – 11 \ge 13y + 19$

The LHS of the inequality is $7y – 11$

The RHS of the inequality is $13y + 19$

The inequality means $7y – 11$ is greater than or equal to $13y + 19$.

Types of Linear Inequalities Based on Number of Inequality Signs

Based on the number of inequality signs there are two types of linear inequalities. These are as follows.

• Simple Inequalities
• Compound Inequalities

Simple Inequalities

An inequality is said to be an open inequality if it has only one inequality sign.

Examples of Simple Inequalities

Example 1: $3x \lt 9$

In $3x \lt 9$, there is only one inequality sign, i.e., $\lt$, hence it is an open inequality.

Example 2: $2x + 5 \ge 6x – 9$

In $2x + 5 \ge 6x – 9$, there is only one inequality sign, i.e., $\ge$, hence it is an open inequality.

Compound Inequalities

An inequality is said to be a compound inequality if it has more than one (usually two) inequality sign.

Examples of Compound Inequalities

Example 1: $2 \lt 3x – 7 \lt 15$

$2 \lt 3x – 7 \lt 15$, there are two inequality signs, first $\lt$ before $3x – 7$ and the other $\lt$ after $3x – 7$.

Example 2: $-7 \le 2x + 5 \gt 18$

$-7 \le 2x + 5 \gt 18$, there are two inequality signs, first $\le$ before $2x + 5$ and the second is $\gt$ after $2x + 5$.

Types of Linear Inequalities Based on Type of Inequality Sign

Based on the type of inequality signs there are two types of linear inequalities. These are as follows.

• Strict Inequalities
• Slack Inequalities

Strict Inequalities

The strict inequalities use symbols like less than ($\lt$) or greater than ($\gt$). These two symbols are called strict inequalities, as it shows the numbers are strictly greater than or less than each other.

Examples of Strict Inequalities

Example 1: $7 \lt 12$ ( $7$ is strictly less than $12$)

Example 2: $m \gt 18$ ($m$ is strictly greater than $18$)

Slack Inequalities

The slack inequalities use symbols like less than or equal to ($le$) or greater than or equal to ($\ge$). The slack inequalities represent the relation between two inequalities that are not strict.

Examples of Slack Inequalities

Example 1: $x \ge 21$ ($x$ is greater than or equal to $21$)

Example 2: $u \le 11$ ($u$ is less than or equal to $11$)

Inequalities Symbols Used in Linear Inequalities

The meaning of inequality is to say that two things are not equal. One of the things may be less than, greater than, less than, equal to, or greater than or equal to the other things. The following are the $5$ different inequality symbols used in linear inequalities.

• $a \ne b$ means that $a$ is not equal to $b$
• $a \lt b$ means that $a$ is less than $b$
• $a \gt b$ means that $a$ is greater than $b$
• $a \le b$ means that $a$ is less than or equal to $b$
• $a \ge b$ means that $a$ is greater than or equal to $b$

Properties of Linear Inequalities

The following are the important properties of linear inequalities.

Transitive Property

The relation between the three numbers is defined using the transitive property. The transitive property of inequalities states that if $a$, $b$, and $c$ are the three numbers, then if $a \ge b$, and $b \ge c$, then $a \ge c$.

Similarly, if $a \le b$, and $b \le c$, then $a \le c$.

Note: If one relation is defined by strict inequality, then the result should also be in strict inequality.

Examples of Transitive Inequalities

Example 1: $15 \gt 12$ and $12 \gt 7$, then $15 \gt 7$

Example 2: $5 \lt 17$ and $17 \lt 36$, then $5 \lt 36$

Addition and Subtraction Property

The addition and subtraction property of inequalities states that adding or subtracting the same constant on both sides of inequalities are equivalent to each other.

For a constant $k$, if $a \lt b$, then $a + k \lt b + k$ and if $a \gt b$, then $a + k \gt b + k$

Similarly, if $a \lt b$, then $a – k \lt y – k$ and if $a \gt b$, then $a – k \gt b – k$

Examples of Addition and Subtraction Property

Example 1: $17 \lt 53$, then $17 + 6 \lt 53 + 6$, i.e., $23 \lt 70$

Example 2: $42 \gt 18$, then $42 + 8 \gt 18 + 8$, i.e., $50 \gt 26$

Example 3: $19 \lt 21$, then $19 – 4 \lt 21 – 4$, i.e., $15 \lt 17$

Example 4: $42 \gt 18$, then $42 – 4 \gt 18 – 4$, i.e., $38 \gt 14$

Multiplication and Division Property

If a positive constant number is multiplied or divided by both sides of an inequality, the inequality remains the same. But, if inequality is multiplied or divided by the negative constant number, the inequality expression will get reversed.

Let $k$ be a positive constant, if $x \lt y$, then $kx \lt ky$ (if $k \gt 0$)

Similarly if $x \gt y$, then $kx \gt ky$ (if $k \gt 0$)

Let $k$ be a negative constant number, if $x \lt y$, then $kx \gt ky$ (if $k \lt 0$)

Similarly if $x \gt y$, then $kx \lt ky$ (if $k \lt 0$)

Note: The above condition holds true for the division operation.

Examples of Multiplication and Division Property

Example 1: $2 \lt 7$, then $5 \times 2 \lt 5 \times 7$, i.e., $10 \lt 35$

Example 2: $9 \gt 3$, then $3 \times 9 \gt 3 \times 3$, i.e., $27 \lt 15$

Example 3: $5 \lt 11$, then $-2 \times 5 \gt -2 \times 11$, i.e., $-10 \gt -22$

Example 4: $13 \gt 7$, then $-4 \times 13 \lt -4 \times 7$, i.e., $-52 \lt -28$

Converse Property

The converse property states that if we flip the number, we have to flip the inequality symbol also, i.e., if $a \gt b$, then $b \lt a$

Similarly if $a \lt b$, then $b \gt a$.

Examples of Closure Property

Example 1: $5 \lt 9$, then $9 \gt 5$

Example 2: $15 \gt 6$, then $6 \lt 15$

Solving Inequalities

Solving inequalities is very much similar to solving an equation. While solving the inequalities, we follow the below-mentioned rules, which do not affect the inequality direction:

• Add or subtract the same number on both sides of an inequality.
• Multiply or divide the inequality by the same positive number.
• Simplify a side of the inequality.

Examples on Solving Inequalities

Example 1: $12x + 3 \lt 8x + 15$

$12x + 3 \lt 8x + 15$

Subtracting $3$ on both sides

$12x + 3 – 3 \lt 8x + 15 – 3$

$=> 12x \lt 8x + 12$

Subtracting $8x$ on both sides

$12x – 8x \lt 8x + 12 – 8x$

$=> 4x \lt 12$

Dividing both sides by $4$

$\frac{4x}{4} \lt \frac{12}{4}$

$x \lt 3$

Example 2: $7x + 9 \lt 9x + 12$

$7x + 9 \lt 9x + 12$

Subtracting $9$ on both sides

$7x + 9 – 9 \lt 9x + 12 – 9$

$=> 7x \lt 9x + 3$

Subtracting $9x$ on both sides

$7x – 9x \lt 9x + 3 – 9x$

$=> -2x \lt 3$

Key Takeaways

The following are the important points to remember about linear inequalities.

• In the case of linear inequalities, some other relationship like less than or greater than exists between LHS and RHS.
• A linear inequality is called so due to the highest power(exponents) of the variable being $1$.
• “Less than” and “greater than” are strict inequalities while “less than or equal to” and “greater than or equal to” are not strictly linear inequalities.
• For every linear inequality which uses strict linear inequality, the value obtained for $x$ is shown by a hollow dot. It shows that the value obtained is excluded.
• For every linear inequality which is not strict inequality, the value obtained for $x$ is shown by a solid dot. It shows that the value obtained is included.

Practice Problems

1. Solve the following inequalities
• $3(x – 1) \le 2 (x – 3)$