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

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

Question
Piecewise-Defined Functions
asked 2021-02-22
Solve the absolute value inequality.
\(\displaystyle{8}{<}{\left|{7}-{3}{x}\right|}\)</span>

Answers (1)

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 \(\displaystyle{8}{<}{\left|{7}-{3}{x}\right|}\ {i}{s}{x}{<}-{\frac{{{1}}}{{{3}}}}\ {\quad\text{or}\quad}\ {x}{>}{5}\).
0

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