Question

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

Equations and inequalities
ANSWERED
asked 2021-03-26
Solve the equation. Write the solution sets to the inequalities in interval notation.
\(\displaystyle{\left|{5}{x}-{1}\right|}={\left|{3}-{4}{x}\right|}\)

Answers (1)

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 \(\displaystyle{x}={\frac{{{9}}}{{{4}}}}\)

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