Solving linear inequations — lesson. Mathematics State Board, Class 7.

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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

$2$

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

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

$2$ .

$x$ 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.

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

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

$2 x + 10 < 70$ , where

$x$ is a natural number.

Step 1) Subtract 10 or add

$-$ 10 on both sides. Hence balance is not disturbed.

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

Step 2) Divide both sides by 2.

$\begin{array}{l} \frac{2 x}{2} = \frac{60}{2} \\ x = 30 \end{array}$

Therefore

$x =$ 30.

$1$

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2. Solving linear inequations

Theory: