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
- Solve the following inequalities
- $3(x – 1) \le 2 (x – 3)$
- $\frac{x – 2}{x + 5}$ \gt 2$
- $\frac{3x – 4}{2} \ge \frac{x + 1}{3} – 1$
- $5(2x – 7) – 3(2x + 3) \le 0$
- $2x + 19 \le 6x + 47$
- The longest side of a triangle is $3$ times the shortest side, and the third side is $2$ cm shorter than the longest side. If the perimeter of the triangle is at least $61$ cm, find the minimum length of the shortest side.
- A manufacturer has $600$ liters of a $12\%$ solution of acid. How many liters of a $30\%$ acid solution must be added to it so that the acid content in the resulting mixture will be more than $15\%$ but less than $18\%$?
FAQs
What are inequalities in math?
When two or more algebraic expressions are compared using the symbols $\lt$, $\gt$, $\le$, or $\ge$, then they form an inequality. They are mathematical expressions in which both sides are not equal.
What is the difference between equations and inequalities?
The following are the differences between equations and inequalities.
a) Equations have $=$ symbol in them, whereas inequalities have $\gt$, $\lt$, $\ge$, or $\le$ in them.
b) The number of solutions of an equation depends on the degree of the equation, whereas an inequality may have a single, unique, or no solution. It doesn’t depend on the degree.
c) By applying any operation on both sides, an equation still holds. If we multiply/divide both sides of an inequality by a negative number, the sign changes.
What are the 5 inequality symbols?
The 5 inequality symbols are less than ($\lt$), greater than ($\gt$), less than or equal ($\le$), greater than or equal ($\ge$), and the not equal symbol ($ne$).
How do you tell whether it’s an inequality?
Equations and inequalities are mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are supposed to be equal and shown by the symbol $=$. Whereas in inequality, the two expressions are not necessarily equal and are indicated by the symbols $\gt$, $\lt$, $\ge$ or $\le$.
Conclusion
Inequalities are the mathematical statement that shows the relation between two expressions using the inequality symbol. The five inequality symbols used in inequalities are $\ne$, $\lt$, $\gt$, $\le$, and $\ge$. Solving inequalities is similar to that of solving equations with one main difference and it is that whenever an inequality is multiplied or divided by a negative number, the inequality sign changes.
Recommended Reading
- What is the Meaning of Equation – Definition, Types & Examples
- How to Solve Linear Equations with Matrices(With Method & Examples)
- Linear Equations in Two Variables – Definition, Types, and Graphs
- Pair of Linear Equations in Two Variables(With Methods & Examples)
- Linear Equations in One Variable – Graph & Method of Solving
