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,
$$\Rightarrow 6x=9$$
$$\Rightarrow x={\displaystyle \frac{9}{6}}$$
$$\therefore x={\displaystyle \frac{3}{2}}$$

Hence, for the given equation $$\left|3x-4\right|=\left|3x-5\right|$$ , the value of $$x={\displaystyle \frac{3}{2}}$$.

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.