# Solve the absolute value inequality. 8<|7-3x|

Question
Piecewise-Defined Functions
Solve the absolute value inequality.
$$\displaystyle{8}{<}{\left|{7}-{3}{x}\right|}$$</span>

2021-02-23
Step 1
First isolate the absolute value expression to the left side of inequality to solve the inequality. Remove the absolute bars by setting a compound inequality. Then solve the inequalities to obtain the solution.
Step 2
The solution to the inequality $$\displaystyle{8}{<}{\left|{7}-{3}{x}\right|}$$</span> can be found as follows,
First isolate the absolute expression by changing the sides as $$\displaystyle{\left|{7}-{3}{x}\right|}{>}{8}$$.
Now apply absolute rule: if $$\displaystyle{\left|{u}\right|}{>}{a}$$, then $$\displaystyle{u}{<}-{a}\ {\quad\text{or}\quad}\ {u}{>}{a}$$ to remove the absolute bars and solve the two inequalities,
$$\displaystyle{\left|{7}-{3}{x}\right|}{>}{8}$$ then
$$\displaystyle{7}-{3}{x}{<}-{8}\ {\quad\text{or}\quad}\ {7}-{3}{x}{>}{8}$$
$$\displaystyle{3}{x}{>}{15}\ {\quad\text{or}\quad}\ {3}{x}{<}-{1}$$</span>
$$\displaystyle{x}{>}{5}{\quad\text{or}\quad}{x}{<}-{\frac{{{1}}}{{{3}}}}$$</span>
Hence, the solution to the absolute value inequality $$\displaystyle{8}{<}{\left|{7}-{3}{x}\right|}\ {i}{s}{x}{<}-{\frac{{{1}}}{{{3}}}}\ {\quad\text{or}\quad}\ {x}{>}{5}$$.

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