# Solve the equation. Write the solution sets to the inequalities in interval notation. |5x-1|=|3-4x|

Equations and inequalities
Solve the equation. Write the solution sets to the inequalities in interval notation.
$$\displaystyle{\left|{5}{x}-{1}\right|}={\left|{3}-{4}{x}\right|}$$

2021-03-28

Solve the given equation as shown, follow the steps below:
$$\displaystyle{\left|{5}{x}-{1}\right|}={\left|{3}-{4}{x}\right|}{\left[{r}{e}{c}{a}{l}{l}:\ {\left|{a}\right|}={\left|{b}\right|}\Rightarrow{a}=\pm{b}\right]}$$
$$\displaystyle\Rightarrow{5}{x}-{1}=\pm{\left({3}-{4}{x}\right)}$$ [add 1 in both sides]
$$\displaystyle\Rightarrow{5}{x}={1}\pm{\left({3}-{4}{x}\right)}$$
Solve the two equations that occured.
$$\displaystyle{5}{x}={1}-{\left({3}-{4}{x}\right)}$$
$$\displaystyle\Rightarrow{5}{x}={1}-{3}+{4}{x}$$
$$\displaystyle\Rightarrow{5}{x}-{4}{x}=-{2}$$
$$\displaystyle\Rightarrow{x}=-{2}$$
$$\displaystyle{5}{x}={1}+{\left({3}-{4}{x}\right)}$$
$$\displaystyle\Rightarrow{5}{x}={1}+{3}-{4}{x}$$
$$\displaystyle\Rightarrow{5}{x}+{4}{x}={1}+{3}$$
$$\displaystyle\Rightarrow{9}{x}={4}$$
$$\displaystyle\Rightarrow{x}={\frac{{{9}}}{{{4}}}}$$
The solutions of this equation are the x=-2 and $$\displaystyle{x}={\frac{{{9}}}{{{4}}}}$$