Question

If $|x - 2|\, = 9$, which of the following would equal $|x + 3|$?
A. 4
B. 7
C. 8
D. 10
E. 11

Answer 1

Here the algebraic equation is a combination of constant and variables. We have to solve the given equation for the variable $x$. Since the equation involves the modulus, by using the definition of modulus or absolute value and simple arithmetic operation we determine the value of $x$. Then we substitute the value of $x$ in another modulus function and hence we obtain the answer.

Complete step by step answer:
The absolute value or modulus of a real function f(x), it is denoted as |f(x)|, is the non-negative value of f(x) without considering its sign. The value of |f(x)| defined as,

& +f(x);\,\,f(x)\ge 0 \\\ & -f(x);\,\,f(x)\le 0 \\\ \end{aligned} \right.$$ Now consider the given question $$|x - 2|\, = 9$$. By the definition of absolute number we are determined the unknown variable and by definition the modulus, separate $$|x - 2|\, = 9$$ into two equations: $$x - 2 = 9$$................(1) And $$ - \left( {x - 2} \right) = 9$$..................(2) Consider the equation (1) $$ \Rightarrow \,\,\,x - 2 = 9$$ Add both side by 2, then $$ \Rightarrow \,\,\,x - 2 + 2 = 9 + 2$$ On simplification, we get $$\therefore \,\,\,x = 11$$ Now consider the equation (2) $$ \Rightarrow \,\,\, - \left( {x - 2} \right) = 9$$ First multiply the -ve sign inside to the parenthesis on LHS. $$ \Rightarrow \,\,\, - x + 2 = 9$$ Add -2 on both side, then $$ \Rightarrow \,\,\, - x + 2 - 2 = 9 - 2$$ On simplification, we get $$ \Rightarrow \,\,\, - x = 7$$ Multiply or Cancel – ve sign on both side $$\therefore \,\,\,x = - 7$$ Hence, the value of x in the equation $$|x - 2|\, = 9$$ is 14 and -7. Now we will substitute these values in the second function i.e., $$|x + 3|$$ When x is 14 $$ \Rightarrow |14 + 3| = 17$$ When x is -7 $$ \therefore | - 7 + 3| = | - 4| = 4$$ In the given options the 17 is not there but 4 is there. **Therefore the option A is the correct one.** **Note:** The algebraic equation or an expression is a combination of variables and constants, it also contains the coefficient. The alphabets are known as variables. The $x, y, z$ etc., are called variables. The numerals are known as constants. The numeral of a variable is known as co-efficient. we must know about the modulus definition.