Question

Solve the equation: \[\left| {3x - 4} \right| = \left| {3x - 5} \right|\]?

Answer 1

Hint: The absolute value of something, we put "|" marks either side (they are called "bars"). The absolute value means to remove any negative sign in front of a number, and to think of all numbers as positive (or zero). Here we have an equation involving absolute values. As a general rule \[\left| a \right| = + a\] and \[\left| a \right| = - a\]. This rule will be applied to RHS. We will solve the given equation in this way to get the final output.

Complete step by step answer:
Given that, \[\left| {3x - 4} \right| = \left| {3x - 5} \right|\]. Now we will apply the general rule of absolute values.In first case, when \[\left| a \right| = + a\] , then we will get,
\[ \Rightarrow 3x - 4 = 3x - 5\]
By transposing method, we will move the LHS term to RHS, we will get,
\[ \Rightarrow - 4 = 3x - 5 + 3x\]
On evaluating this, we will get,
\[ \Rightarrow - 4 = - 5\] which is not possible.
From this first case, we will conclude that, there are no solutions for \[\left| a \right| = + a\].

In second case, when \[\left| a \right| = - a\] , then we will get,
\[ \Rightarrow 3x - 4 = - (3x - 5)\]
Removing the brackets, we will get,
\[ \Rightarrow 3x - 4 = - 3x + 5\]
Again by using the transposition method, we will move the RHS term i.e. 3x to LHS, we will get,
\[ \Rightarrow 3x + 3x = 5 + 4\]
On evaluating this, we will get,
\[ \Rightarrow 6x = 9\]
\[ \Rightarrow x = \dfrac{9}{6}\]
\[ \therefore x = \dfrac{3}{2}\]

Hence, for the given equation \[\left| {3x - 4} \right| = \left| {3x - 5} \right|\] , the value of \[x = \dfrac{3}{2}\].

Note: Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative. The number line is not just a way to show distance from zero; it's also a useful way to graph equalities and inequalities that contain expressions with absolute value.