Question

How do you solve \[|x - 7|\, = 4\]?

Answer 1

Hint: Here the equation is an algebraic equation that is a combination of constant and variables. We have to solve the given equation for 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.

Complete step-by-step solution:
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 \[|f(x)| = \left\{
  \begin{array}{*{20}{c}}
  { + f(x)\,;}&{f(x) \geqslant 0}
\end{array} \\
  \begin{array}{*{20}{c}}
  { - f(x)\,;}&{f(x) < 0}
\end{array} \right\}\]
Now consider the given question \[|x - 7|\, = 4\]
By the definition of absolute number we are determined the unknown variable and
By definition the modulus separate \[|x - 7|\, = 4\]into two equations:
\[x - 7 = 4\] (1)
 and
\[ - \left( {x - 7} \right) = 4\] (2)
Consider the equation (1)
\[ \Rightarrow \,\,\,x - 7 = 4\]
Add both side by 7, then
\[ \Rightarrow \,\,\,x - 7 + 7 = 4 + 7\]
On simplification, we get
\[\therefore \,\,\,x = 11\]
Now consider the equation (2)
\[ \Rightarrow \,\,\, - \left( {x - 7} \right) = 4\]
First multiply the -ve sign inside to the parenthesis on LHS.
\[ \Rightarrow \,\,\, - x + 7 = 4\]
Add -7 sign on both side, then
\[ \Rightarrow \,\,\, - x + 7 - 7 = 4 - 7\]
On simplification, we get
\[ \Rightarrow \,\,\, - x = - 3\]
Multiply or Cancel – ve sign on both side
\[\therefore \,\,\,x = 3\]
Hence, the value of x in the equation \[|x - 7|\, = 4\] is 11 and 3.
Verification:
Put x=11 to the equation \[|x - 7|\,\], then
\[ \Rightarrow \,\,\,|11 - 7|\,\]
\[ \Rightarrow \,\,\,|4|\,\]
\[ \Rightarrow \,\,\,4\]
\[\therefore \,\,\,|x - 7|\, = 4\]
And
Put x=3 to the equation \[|x - 7|\,\], then
\[ \Rightarrow \,\,\,|3 - 7|\,\]
\[ \Rightarrow \,\,\,| - 4|\,\]
By the definition of modulus \[| - x|\, = x\], then
\[ \Rightarrow \,\,\,4\]
\[\therefore \,\,\,|x - 7|\, = 4\]
Hence verified.

Note: : The algebraic equation or a expression is a combination of variables and constants, it also contains the co-efficient. The alphabets are known as variables. The x, y, z etc., are called as variables. The numerals are known as constants. The numeral of a variable is known as co-efficient. We have 3 types of algebraic expressions namely monomial expression, binomial expression and polynomial expression. By using the tables of multiplication, we can solve the equation.