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A simple equation has only one solution. But an inequation has many solutions.

Example:

Consider an equation, \( 2x + 4 = 10\). If we simplify this equation, we get \( x = 3\).

If we take the above equation as inequation, the given expression will have many solutions.

Say \(2x + 4 < 10\) where \(x\) is a natural number.

Now we simplify and find the solutions.

**Step**\(- 1\): Subtract \(4\) on both sides.

\(2x + 4 - 4 < 10 - 4\).

\(2x + 0 < 6\).

\(2x < 6\).

**Step**\(- 2\): Divide both sides by \(2\)

$\begin{array}{l}\Rightarrow \frac{2x}{2}<\frac{6}{2}\\ \\ \Rightarrow x<3\end{array}$

Since \(x\) is a natural number, the solutions of \(x\) are less than \(3\).

Thus, the solutions for the inequation \(2x + 4 < 10\) are \(1\), and \(2\).

If \(x\) is an integer, the negative number can also be the solutions of \((x)\).

So far, we have learned about the possible solutions of an inequation; now we understand how to solve the inequation and obtain the solutions.

Click! here to recall on how to solve an equation.

**Rules to solve Inequations:**

To solve an algebraic equation, we generally use the arithmetic operation \(( +, -, ×, ÷ )\). We are going to apply the same method to solve an inequations.

**1**. Addition/Subtraction of the same number on both sides of the inequation does not change the value of the inequation.

Example:

Let's take an inequation \(4 < 8\).

Now we are adding the same number on both sides.

\((4 + 2 < 8 + 2)\) ----------- [Inequation balance is not disturbed]

\(6 < 10\).

As an extension of this result, adding/subtracting any number '\(x\)' instead of \(2\), does not change the inequation ⇒ \(4 + x < 8 + x\).

**2**. Multiplication by the same positive number on both sides of the inequation does not change the inequation value.

Example:

Consider an inequation \(4 < 8\).

\(⇒ (4 × 2 < 8 × 2)\) --------- [Multiply by \(2\) in both sides]

\( ⇒ 8 < 16\).

As an extension of this result, multiplying any positive number '\(x\)' instead of \(2\) does not change the inequation.

\(⇒\) \(4 × x < 8× x\).

**3**. Division by the non-zero same positive number on both sides of the inequation does not change the inequation value.

Example:

Let's consider the inequation \( 4 < 8\). And divide by \(2\) on both sides.

$\begin{array}{l}\Rightarrow \frac{2x}{2}<\frac{6}{2}\\ \\ \Rightarrow x<3\end{array}$

As an extension of this result, dividing any positive number '\(x\)' instead of \(2\) does not change the inequation ⇒ \(4 ÷ x < 8 ÷ x\).

Let us solve an inequation by applying the above rules.

Example:

Using the linear rules, solve the inequation $2x+10<70$, where \(x\) is a natural number.

**Step 1)**Subtract 10 or add \(-\)10 on both sides. Hence balance is not disturbed.

$\begin{array}{l}2x+10-10=70-10\\ \\ 2x+0=60\\ \\ 2x=60\end{array}$

**Step 2)**Divide both sides by 2.

$\begin{array}{l}\frac{2x}{2}=\frac{60}{2}\\ \\ x\phantom{\rule{0.147em}{0ex}}=30\end{array}$

**Therefore**\(x =\) 30.