Linear Equations and Inequalities in One Variable — Mental Maths

Fill in the blanks:

(i) An equation of the type $ax + b = 0$ where $a \ne 0$ is called a ............... in variable $x$.

(ii) Any value of the variable which satisfies the equation is called a ............... of the equation.

(iii) The process of finding all the solutions of an equation is called ...............

(iv) We can add the ............... to both sides of an equation.

(v) We can divide both sides of an equation by the same ............... number.

(vi) The solution set of the inequality $3x \le 10, x \in N$ is ............... .

Answer

(i) An equation of the type $ax + b = 0$ where $a \ne 0$ is called a linear equation in variable $x$.

(ii) Any value of the variable which satisfies the equation is called a solution of the equation.

(iii) The process of finding all the solutions of an equation is called solving the equation.

(iv) We can add the same number to both sides of an equation.

(v) We can divide both sides of an equation by the same non-zero number.

(vi) $3x \le 10$

$\Rightarrow x \le \dfrac{10}{3} \\[1em] \Rightarrow x \le 3.33.....$

∵ $x$ is a natural number.

The solution set of the inequality $3x \le 10, x \in N$ is {1, 2, 3}.

State whether the following statements are true (T) or false (F):

(i) An equation is a statement that two expressions are equal.

(ii) We cannot subtract the same number from both sides of an equation.

(iii) $3x + 2 = 4(x + 7) + 9$ is a linear equation in variable $x$

(iv) $x = 1$ is the solution of equation $4(x + 5) = 24$

Answer

(i) An equation is a statement that two expressions are equal.

This statement is True.

(ii) We cannot subtract the same number from both sides of an equation.

We can subtract the same number from both sides of an equation.

So, this statement is False.

(iii) $3x + 2 = 4(x + 7) + 9$ is a linear equation in variable $x$

On simplifying, we get

$\Rightarrow 3x + 2 = 4x + 28 + 9 \\[1em] \Rightarrow 3x + 2 = 4x + 37 \\[1em] \Rightarrow - x - 35 = 0$

As it is of the type $ax + b = 0$, where $a \ne 0$, it is a linear equation in variable $x$.

So, this statement is True.

(iv) $x = 1$ is the solution of equation $4(x + 5) = 24$

Substituting $x = 1$ in L.H.S., we get

⇒ L.H.S. = 4(1 + 5) = 4 × 6 = 24 = R.H.S.

So, this statement is True.