 UPSKILL MATH PLUS

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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 $$2$$

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

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.

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

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.