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

Hint: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,
\[|f(x)|=\left\{ \begin{align}
  & +f(x);\,\,f(x)\ge 0 \\
 & -f(x);\,\,f(x)\le 0 \\
\end{align} \right.\]
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.